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Preface

The XV Spanish Meeting on Computational Geometry (XV EGC,Encuentros de Geometría Computacional)was held on June 2628, 2013 in Seville, Spain. The EGC meetings were rst held annually from 1990 to 1995, andbiennually since then. The meeting combines a strong scientic tradition with a friendly, collegial atmosphere.The original aim of our meeting was to provide a forum where Spanish-speaking researchers and students couldpresent their research activities. Since then the EGC has evolved; in 2011, the XIV EGC was an internationalgathering for the rst time, and the abstracts contained in the 2011 proceedings were peer reviewed.

This volume contains the four-page extended abstracts of the presentations accepted and delivered at the2013 EGC, and the four invited lectures. A selection of the papers accepted for the EGC will be publishedin the International Journal of Computational Geometry and Applications following a formal review accordingto the standards established by the latter journal. We would like to thank all the authors and attendees fortheir participation in the XV ECG. Special thanks are expressed to the members of the Scientic Committeeand the referees for their careful review of the papers and their insightful comments. Lastly, we are gratefulfor the generous support of our sponsors: Departamento de Matemática Aplicada II, Vicerrectorado de Rela-ciones Institucionales, IMUS (Instituto de Matemáticas) and Departamento de Matemática Aplicada I, all of theUniversidad de Sevilla.

The Editors

Committees

Scientic Committee Organizing Committee

Oswin Aichholzer Carmen CortésSergey Bereg María José ChávezProsenjit K. Bose José Miguel Díaz-Báñez (chair)Sergio Cabello Isabel FernándezJosé Miguel Díaz-Báñez (co-chair) Delia GarijoAlfredo García M. Ángeles GarridoFerran Hurtado Alberto MárquezRolf Klein Pablo Pérez-LanteroStefan Langerman Inmaculada VenturaMario LópezAlberto Márquez (co-chair)Belén PalopPedro RamosDavid RappaportVera SacristánGelasio SalazarJ. Antoni SellarèsJorge Urrutia (co-chair)David Wood

ii

Contents

Invited speaker 1Wednesday 9:3010:30

Art gallery problems, old and recent

Jorge Urrutia 1

Session 1Wednesday 10:3011:30

Continuous surveillance of points by rotating oodlights

Sergey Bereg, José Miguel Díaz-Báñez, Marta Fort, Mario A. López, Pablo Pérez-Lantero,and Jorge Urrutia 3

Some results on open edge guarding of polygons

Antonio L. Bajuelos, Santiago Canales, Gregorio Hernández, Mafalda Martins,and Inês Matos 7

Guarding the vertices of thin orthogonal polygons is NP-hard

Ana Paula Tomás 11


Solving common inuence region queries with the GPU

Marta Fort and J. Antoni Sellarès 15

Reporting ock patterns on the GPU

Marta Fort, J. Antoni Sellarès, and Nacho Valladares 19

Parallel constrained Delaunay triangulation

Narcís Coll and Marité Guerreri 23

Metaheuristic approaches for the Minimum Dilation Triangulation problem

Maria Gisela Dorzán, Mario Guillermo Leguizamón, Efrén Mezura-Montes,and Gregorio Hernández 27

Invited speaker 2Wednesday 15:3016:30

Three location tapas calling for CG sauce

Frank Plastria 31


On the barrier-resilience of arrangements of ray-sensors

David Kirkpatrick, Boting Yang, and Sandra Zilles 35

Computing the stretch of an embedded graph

Sergio Cabello, Markus Chimani, and Petr Hlim¥ný 39

An algorithm that constructs irreducible triangulations of once-punctured surfaces

María José Chávez, Serge Lawrencenko, José R. Portillo, and M. Trinidad Villar 43

iii

On the enumeration of permutominoes

Ana Paula Tomás 47

Distance domination, guarding and vertex cover for maximal outerplanar graphs

Santiago Canales, Gregorio Hernández, Mafalda Martins, and Inês Matos 51

Invited speaker 3Thursday 09:0010:00

Abstract Voronoi diagrams

Rolf Klein 55

Session 4Thursday 10:0011:00

Equipartitioning triangles

Pedro Ramos and William Steiger 57

On the nonexistence of k-reptile simplices in R3 and R4

Jan Kyn£l and Zuzana Safernová 61

Drawing the double circle on a grid of minimum size

Sergey Bereg, Ruy Fabila-Monroy, David Flores-Peñaloza, Mario A. López,and Pablo Pérez-Lantero 65


SensoGraph: Using proximity graphs for sensory analysis

David N. de Miguel, David Orden, Encarnación Fernández-Fernández,José M. Rodríguez-Nogales, and Josena Vila-Crespo 69

Simulated Annealing applied to the MWPT problem

Edilma Olinda Gagliardi, Mario Guillermo Leguizamón, and Gregorio Hernández 73

A symbolic-numeric dynamic geometry environment for the computation of equidistant curves

Miguel A. Abánades and Francisco Botana 77

Simulating distributed algorithms for lattice agents

Oswin Aichholzer, Thomas Hackl, Vera Sacristán, Birgit Vogtenhuber,and Reinhard Wallner 81


Empty convex polytopes in random point sets

József Balogh, Hernán González-Aguilar, and Gelasio Salazar 85

Note on the number of obtuse angles in point sets

Ruy Fabila-Monroy, Clemens Huemer, and Eulàlia Tramuns 89

Stabbing simplices of point sets with k-ats

Javier Cano, Ferran Hurtado, and Jorge Urrutia 91

Stackable tessellations

Lluís Enrique and Rafel Jaume 95

iv

Session 7Friday 09:3011:10

Improved enumeration of simple topological graphs

Jan Kyn£l 99

On three parameters of invisibility graphs

Josef Cibulka, Miroslav Korbelá°, Jan Kyn£l, Viola Mészáros, Rudolf Stola°,and Pavel Valtr 103

On making a graph crossing-critical

César Hernández-Vélez and Jesús Leaños 107

Witness bar visibility

Carmen Cortés, Ferran Hurtado, Alberto Márquez, and Jesús Valenzuela 111

The alternating path problem revisited

Mercè Claverol, Delia Garijo, Ferran Hurtado, Dolores Lara, and Carlos Seara 115

Session 8Friday 11:4013:00

Phase transitions in the Ramsey-Turán theory

József Balogh 119

On 4-connected geometric graphs

Alfredo García, Clemens Huemer, Javier Tejel, and Pavel Valtr 123

Monotone crossing number of complete graphs

Martin Balko, Radoslav Fulek, and Jan Kyn£l 127

Flips in combinatorial pointed pseudo-triangulations with face degree at most four

Oswin Aichholzer, Thomas Hackl, David Orden, Alexander Pilz, Maria Saumell,and Birgit Vogtenhuber 131

Invited speaker 4Friday 13:1514:15

Recent developments on the crossing number of the complete graph

Pedro Ramos 135

v

XV Spanish Meeting on Computational Geometry, June 26-28, 2013


Jorge Urrutia∗

Instituto de Mathemáticas, Universidad Nacional Autónoma de Mexico (UNAM), Mexico City, México.

Abstract

In 1973, Victor Klee posed the following question:How many guards are necessary, and how many aresucient to patrol the paintings and works of art in anart gallery with n walls? This wonderfully naïve ques-tion of combinatorial geometry has, since its formula-tion, stimulated a plethora of papers, surveys and abooks. The rst result in this area, due to V. Chvátal,asserts that ⌊n

3 ⌋ guards are occasionally necessary andalways sucient to guard an art gallery representedby a simple polygon with n vertices. Since Chvátal'sresult, numerous variations on the art gallery problemhave been studied, including mobile guards, guardswith limited visibility or mobility, illumination of fam-ilies of convex sets on the plane, guarding of rectilin-ear polygons, and others. In this talk, we will reviewsome old results in this area of research, and reviewsome new results on Art Galleries, including some re-cent results in R3, as well as results using rotatingoodlights.

∗Email: [email protected]. Partially supported

by CONACYT of México Grant 178379, and MEC project

MTM2009-08652.

1


2


Continuous surveillance of points by rotating floodlights

S. Bereg∗1, J. M. Díaz-Báñez†2, M. Fort‡3, M. A. Lopez§4, P. Pérez-Lantero¶5, and J. Urrutia‖6

1Department of Computer Science, University of Texas at Dallas, USA.2Departamento Matemática Aplicada II, Universidad de Sevilla, Spain.

3Departament d’Informàtica i Matemàtica Aplicada, Universitat de Girona, Spain.4Department of Computer Science, University of Denver, USA.

5Escuela de Ingeniería Civil en Informática, Universidad de Valparaíso, Chile.6Instituto de Matemáticas, UNAM, Mexico.

Abstract

Let P and F be sets of n ≥ 2 and m ≥ 2 points in theplane, respectively, so that P∪F is in general position.We study the problem of finding the minimum angleα ∈ [2π/m, 2π] such that one can install at each pointof F a stationary rotating floodlight with illuminationangle α, initially oriented in a suitable direction, insuch a way that, at all times, every target point of P isilluminated by at least one light. All floodlights rotateat unit speed and clockwise. We give an upper boundfor the 1-dimensional problem and present results forsome instances of the general problem. Specifically,we solve the problem for the case in which we havetwo floodlights and many points, and give an upperbound for the case in which there are many floodlightsand only two target points.

1 Introduction

Illumination problems are well known in Discrete andComputational Geometry [4]. Let P be a set of n ≥ 2targets and F a set of m ≥ 2 floodlights, both definedin the plane. We assume that P ∪ F is in general po-sition and that all floodlights rotate clockwise at unitspeed. We say that F covers P with illumination an-gle α ≥ 2π/m if there is a suitable initial orientationof each light so that, at all times, each target pointof P is illuminated by at least one floodlight. We

∗Email: [email protected]. Partially supported by projectMEC MTM2009-08652.†Email: [email protected]. Partially supported by project

MEC MTM2009-08652 and ESF EUROCORES programmeEuroGIGA-ComPoSe IP04-MICINN Project EUI-EURC-2011-4306.‡Email: [email protected]. Partially supported by the

Spanish MCI grant TIN2010-20590-C02-02.§Email: [email protected].¶Email: [email protected]. Partially supported by project

CONICYT, FONDECYT/Iniciación 11110069 (Chile), andproject MEC MTM2009-08652.‖Email: [email protected]. Partially supported by

project MEC MTM2009-08652.

consider the problem of finding the minimum angleα = α(P, F ) ∈ [2π/m, 2π] and initial orientation ofeach floodlight so that F covers P .Our general problem can be considered a discretiza-

tion of the one studied by Kranakis et al. [2]. Giventhe locations of m floodlights (i.e. antennae) and a re-gion, their problem asks to schedule the lights so thatthe entire region is covered at all times. The schedul-ing of static floodlights for covering a given region wasfirst considered by Bose et al. [1]. Research related toour problem can be found in a number of domains, in-cluding art gallery and related problems, multi-targettracking, and multi-robot surveillance tasks [3, 4]. Acomplete review of these fields can be found in [4].We present results for some cases of our general

problem: the elements of P ∪F are located on a givenline (Section 2), the two-dimensional version with twofloodlights (Section 3), and the problem in the planewith two target points (Section 4).Given points u, v in the plane, `(u, v) denotes the

line containing both u and v. We identify any flood-light by the point where it is installed. For any flood-light f , at any instance of time, the region illuminatedby f is delimited by rays f− and f+, starting at f−and ending at f+ in the clockwise direction. We saythat we configure f with angle β if the angle betweenf+ and the positive x-axis is equal to β. Given thatall floodlights rotate at the same speed, it suffices toconsider only the interval of time [0, 2π).

2 Points and floodlights on a line

We first consider the case in which the points of Pand the floodlights of F lie on a line L, say the x-axis. Kranakis et al. [2] considered the case in whichthe floodlights are located on a line. They showed thatthe entire line can be illuminated bym rotating flood-lights using illumination angle 3π/m and this boundis tight. This can be viewed as a special case of ourproblem where n ≥ m+ 1 and each of the m+ 1 seg-ments determined by F contains at least one point of

3


P . We consider other cases and show that the illumi-nation angle is smaller than 3π/m for some of them.Partition P into k − 1 (k ≥ 2) maximal inter-

vals s1, s2, . . . , sk−1, from left to right, each of whichcontains elements of P but no elements of F . LetF1 denote the elements of F to the left of s1, Fi(i = 2, . . . , k−1), the elements of F between si−1 andsi, and Fk, the elements of F to the right of sk−1. Letmi = |Fi| for i = 1, . . . , k. Observe that m1,mk ≥ 0,mi ≥ 1 for i = 2, . . . k−1, andm1+m2+. . .+mk = m.

Lemma 1 Two floodlights f1 and f2 with illumina-tion angle α, belonging to the same set among F1 ∪Fk, F2, F3, . . . , Fk−1, can be configured to cover P dur-ing 2α time in total. Furthermore, if two floodlightsof F cover P with angle α < 3π/2, then they mustbelong to a same set among F1 ∪Fk, F2, F3, . . . , Fk−1.

Proof. Suppose that both f1 and f2 belong to a setFi (i = 1, . . . , k), and assume w.l.o.g. that f1 is to theleft of f2. Configure f1 with angle zero and f2 withangle π (see top of Figure 1a). At time t = π theconfiguration of f1 and f2 is as shown in the bottomof Figure 1a. Since there is no element of P in thesegment connecting f1 and f2 then all elements of Pare illuminated during intervals [0, α] and [π, π + α],2α time in total.

f1 f2

f1 f2

(a)

f1 f2

f1 f2

(b)

Figure 1: Proof of Lemma 1: (a) Case where f1 and f2belong a same set Fi. (b) Case where f1 ∈ F1 and f2 ∈ Fk.

Suppose now that f1 ∈ F1 and f2 ∈ Fk. Configureboth f1 and f2 with angle zero (see top of Figure 1b).At time t = π the configuration of f1 and f2 is asshown in the bottom of Figure 1b. Since all elementsof P belong to the segment connecting f1 and f2 thenall elements of P are illuminated during intervals [0, α]and [π, π + α], 2α time in total.For the second part of the lemma, let f and g be

the two lights that cover P . Clearly α ≥ π. Assume,w.l.o.g., that g 6∈ F1 ∪ Fk and that f is to the left ofg. Also, let Q ⊆ P denote the set of targets to the

right of f . Light f alone can cover Q for an intervalof length α, while g alone can cover Q for α− π only.Since α < 3π/2, an interval of length 2π − α > π/2is unaccounted for by f , but g can pick up at mostα−π < π/2 of this. Hence, f and g must belong to thesame set. The claim does not hold if α ≥ 3π/2.

Lemma 2 Three floodlights f1, f2, and f3 with illu-mination angle α < π, can be configured so that, to-gether, they cover the whole line L (hence, P ) during2α time in total.

Proof. The proof can be obtained from [2]. Assume,w.l.o.g., that f1, f2, and f3 appear in this order fromleft to right. Configure f1, f2, and f3 with angle zero,π, and zero, respectively (top of Figure 2). At timet = π the configuration is as shown at the bottom ofFigure 2. During interval [0, α] the line is illuminatedby f1 and f2, and during interval [π, π+α], by f2 andf3. The result follows.

f1 f2 f3

f1 f2 f3

Figure 2: Proof of Lemma 2

Theorem 3 If all floodlights belong to the same setamong F1 ∪ Fk, F2, F3, . . . , Fk−1, then α(P, F ) =2π/m, which is optimal. Otherwise, α(P, F ) satisfies:

α(P, F ) ≤ min

3π

m,

2π

m−Q+ 2bQ3 c

(1)

where Q denotes the number of odd numbers in theset m1 +mk,m2, . . . ,mk−1.

Proof. Obviously α(P, F ) ≥ 2π/m in all cases. Alllights belong to the same set iff k = 2, or k = 3and m1 = m3 = 0. In both cases, illumination an-gle α = 2π/m is sufficient. Assume first that k = 2and let F1 = f1, . . . , fm1

and F2 = f ′1, . . . , f ′m2.

Floodlight fi is configured with angle (i − 1)α fori = 1, . . . ,m1, and floodlight f ′j , with angle π − jαfor j = 1, . . . ,m2 (see Figure 3). Then, at any timet ∈ [0,m1α) P is covered by a member of F1 and,at any time t ∈ [m1α, 2π), by a member of F2. As-sume now that k = 3 and m1 = m3 = 0. Then,F = F2 = f1, . . . , fm. By configuring fi with angle2iπ/m, P is covered by F . This proves the optimalresult when all lights belong to the same set.

4


f1 f3p1 p2 p3 f4 f5f2

Figure 3: Two groups of floodlights F1 = f1, f2 andF2 = f ′1, f ′2, f ′3 = f3, f4, f5, where m1 = 2 and m2 =3, and the configuration with angle α = 2π/5.

If neither of the two cases above occurs, we con-figure the floodlights to satisfy inequality (1) as fol-lows. We consider the case m1 = mk = 0 and mi = 1(i = 2, . . . , k−1) separately because the result followsimmediately from [2], since our problem is equivalentto illuminating the whole x-axis. In this case m = Qand α(P, F ) = 3π/m = min 3πm ,

2πm−Q+2bQ3 c

which isalso optimal.For the remaining cases, we can only obtain

an upper bound on the optimal illumination an-gle. We proceed as follows. Pair the elements ofF into N = bm1+mk

2 c + bm2

2 c + . . . + bmk−1

2 c =m−Q

2 pairs (f1,1, f1,2), (f2,1, f2,2), . . . , (fN,1, fN,2)so that the elements of each pair belong to asame set among F1 ∪ Fk, F2, F3, . . . , Fk−1. Groupthe remaining Q floodlights into M = bQ3 ctriples (f ′1,1, f

′1,2, f

′1,3), . . . , (f

′M,1, f

′M,2, f

′M,3) (leaving

at most two ungrouped). Let α = 2πm−Q+2bQ3 c

=

2π2N+2M . We now schedule the floodlights as fol-lows. Configure fi,1 and fi,2 with angles (i− 1)α andπ + (i − 1)α, respectively, for i = 1, . . . , N ; and con-figure f ′j,1, f ′j,2, and f ′j,3 with angles (N + j − 1)α,(N + j − 1)α, and π + (N + j − 1)α, respectively, forj = 1, . . . ,M . Finally, arbitrarily configure the re-maining floodlights (at most two). The correctness ofthis configuration follows from lemmas 1 and 2.

3 Many points and two lightsIn this section we consider the case of two floodlightsf1 and f2, i.e., m = 2. Let p1, . . . , pn denote the ele-ments of P . Assume w.l.o.g. that line `(f1, f2) is hor-izontal and that f1 is located to the left of f2. Givenany target point pi (i = 1, . . . , n), let θi ∈ [0, π) de-note the angle at pi in the triangle 4pif1f2. If thereare points from P on both sides of the line `(f1, f2),then we define two angles β+ and β− as the maximumof θi over all points pi above and below `(f1, f2), re-spectively (see Figure 4a). Otherwise, all the points ofP are on the same side of `(f1, f2) and we define twoangles, βmax and βmin, as the largest and smallest θiover all points pi, respectively (see Figure 4b).

Theorem 4 (Two floodlights) For m = 2, n ≥ 2:(1) If there are points of P on both sides of `(f1, f2)then α(P, F ) = π + β++β−

2 .(2) If all the points of P lie on one side of `(f1, f2)then α(P, F ) = π + βmax−βmin

2 .

f1 f2

β+

β−

(a)f1 f2

βmax

βmin

(b)

Figure 4: (a) β+ and β−. (b) βmax and βmin.

Proof. First, we prove part (1) of the theorem. Weconfigure floodlights f1 and f2 initially as follows. LetAf1Bf2 be the quadrilateral such that angle ∠f1Af2is equal to β+, angle ∠f2Bf1 is equal to β−, andpoints f1 and f2 are symmetric with respect to theline `(A,B) as shown in Figure 5a.

f1 f2

β−

β+

A

B

(a)

f1 f2

A

B

β+

β−

(b)

f1

f2

B

(c)

Figure 5: (a) Initial position. (b),(c) General position.

Since f1 and f2 rotate at unit speed, it always holdsthat ∠f1Af2 = β+ and ∠f2Bf1 = β− (see Figure 5b).Furthermore, the region not illuminated by the lightsis always a subset of the interior of the union of the tri-angles 4f1Af2 and 4f2Bf1 (see Figures 5b and 5c),and it never contains points of P by the definition ofβ+ and β−. It remains to show that any illuminationangle smaller than π + β++β−

2 is not feasible.Suppose that, initially, floodlight fi, i = 1, 2 covers

angles in the interval [αi, βi]. First we show that theseintervals cover all possible directions in [0, 2π]. If, tothe contrary, a direction t is not covered by [α1, β1]∪[α2, β2], then there is a rotation such that pi is notilluminated, where pi is the point above the x-axisand ∠f2pif1 = β+, a contradiction. Therefore, the

5


floodlight intervals overlap as shown in Figure 6a.

f1f2

α1

β1 α2

β2

(a)

f1 f2

β+

pi

(b)

Figure 6: The floodlight intervals.

We show now that the overlapping interval [α2, β1]has length at least β+. Suppose to the contrary that itis smaller than β+. Consider the rotation by the angleγ such that the β2-ray of floodlight f2 passes throughthe point pi defining β+ (see Figure 6b). Then theα1-ray of floodlight f1 will not cover pi since the an-gle between the two rays is less than β+. Therefore piis not covered if the rotation angle is slightly smallerthan γ, a contradiction. Similarly, the overlappinginterval [α1, β2] has length at least β−. If the illumi-nation angle is α then 2α ≥ 2π + β+ + βi. The claimof part (1) follows.To prove part (2) of the theorem we configure flood-

lights f1 and f2 at the beginning as follows. LetAf1Bf2 be a quadrilateral such that ∠f1Af2 = βminand ∠f2Bf1 = βmax and points f1 and f2 are sym-metric about line `(A,B) as shown in Figure 7a. Theargument is similar to the proof of part (1) since thepoints A and B move along arcs shown in Figure 7a.The uncovered part is either a region below Cf1Bf2D,shown in Figure 7a, or the wedge XY Z, shown inFigure 7c. In any case the area between the two arcsdefined by βmin and βmax is always illuminated.The optimality of angle π+ βmax−βmin

2 can be shownsimilarly to the proof of part (1).

4 Many lights and two pointsLet p1 and p2 denote the elements of P and f1, . . . , fmdenote the elements of F . Let θi (i = 1, . . . ,m) de-note the angle by which line `(fi, p1) has to be rotatedclockwise with center fi to become line `(fi, p2). As-sume w.l.o.g. that θ1 ≤ θ2 ≤ . . . ≤ θm.

Lemma 5 If n = 2 then α(P, F ) ≤ 2πm + θm−θ1

m .

Proof. It suffices to prove that for α = 2πm + θm−θ1

mthe m floodlights can be configured properly. Con-figure f1 arbitrarily and, for i = 1, . . . ,m − 1, con-figure fi+1 to start illuminating point p2 at the timefi stops illuminating it. Since α ≥ 2π/m then p2 is

f1 f2

A

B

C D

(a)

f1 f2

(b)

f1 f2

X Y

Z

(c)

Figure 7: The floodlight intervals.

always illuminated. Observe that p1 is illuminatedat some time by both fi and fi+1 since θi ≤ θi+1.Then the time spam in which p1 is illuminated byfi and not by fi+1 is equal to α + θi − θi+1. Since∑m−1i=1 (α+ θi − θi+1) + α = mα+ θ1 − θm = 2π then

p1 is always illuminated.

One can build examples, as the following one, in whichθ1 < θm and α is the theoretical minimum (i.e. α =2π/m), showing that α = 2π

m + θm−θ1m is not always

optimal. Let m = 8 and (θ1, θ2, θ3, θ4, θ5, θ6, θ7, θ8) =(π4 ,

π4 ,

π2 ,

π2 ,

π2 ,

π2 ,

3π4 ,

3π4 ). Then f1, . . . , f8 can be con-

figured with α = 2π/m so that p1 is illuminated inthe (circular) order f1, f3, f8, f2, f4, f5, f6, f7 and p2in the order f6, f1, f7, f3, f2, f8, f4, f5. Thus, it be-comes interesting to decide whether α(P, F ) = 2π/mwhen θ1, . . . , θm are all multiples of 2π/m.Acknowledgements. The problem studied here was in-troduced and partially solved during the VI SpanishWork-shop on Geometric Optimization, June 2012, El Rocío,Huelva, Spain. The authors would like to thank otherparticipants for helpful comments.

References[1] J. Bose, L. Guibas, A. Lubiw, M. Overmars, D. Sou-

vaine, and J. Urrutia. The floodlight illuminationproblem. Int. J. Comp. Geom., 7:153–163, 1997.

[2] E. Kranakis, F. MacQuarie, O. Morales, and J. Urru-tia. Uninterrupted coverage of a planar region with ro-tating directional antennae. In ADHOC-NOW, pages56–68, 2012.

[3] K. Sugihara, I. Suzuki, and M. Yamashita. Thesearchlight scheduling problem. SIAM J. Comput.,19(6):1024–1040, 1990.

[4] J. Urrutia. Art gallery and illumination problems. InHdbk. of Comp. Geom., pages 973–1027, 2000.

6


Some results on open edge guarding of polygons

Antonio L. Bajuelos∗1, Santiago Canales2, Gregorio Hernández†3, Mafalda Martins∗‡1, and Inês Matos∗§1

1Universidade de Aveiro & CIDMA, Portugal2Universidad Pontificia Comillas de Madrid, Spain

3Universidad Politécnica de Madrid, Spain

Abstract

This paper focuses on a variation of the Art Galleryproblem that considers open edge guards. The “open”prefix means the endpoints of an edge where a guardis are not taken into account for visibility purposes.This paper studies the number of open edge guardsthat are sufficient and sometimes necessary to guardsome classes of simple polygons.

Introduction

The well known Art Gallery problem studies the mini-mum number of guards that are needed to fully cover apolygon P , that is, the number of guards from whichevery point of P is visible. Ideally, guards may beplaced anywhere on P but usually they are restrictedto vertices of the polygon or its edges. In the first casesuch guards are called vertex guards and in the sec-ond edge guards. Moreover, a point guard is a guardthat can be placed anywhere on the polygon. Lee etal. [3] proved that finding the minimum number ofguards to fully cover a polygon without holes is NP-hard for all three variations of guards. Toussaint con-jectured that bn

4 c edge guards are sufficient to coverany simple polygon of n vertices, except for small val-ues of n. Later, Shermer [4] proved that b 3n

10 c edgeguards are sufficient to cover any simple polygon, ex-cept for n = 3, 6, 13 where an extra edge guard mightbe needed. Shermer actually proved a combinatorialresult: any triangulation of a polygon with n verticescan be dominated by 3n

10 edge guards.In this paper guards are assumed to be placed along

open edges of a polygon, that is, the endpoints of any

∗Research supported by FEDER funds throughCOMPETE–Operational Programme Factors of Com-petitiveness, CIDMA and FCT within project PEst-C/MAT/UI4106/2011 with COMPETE number FCOMP-01-0124-FEDER-022690.†Research supported by ESF EUROCORES programme Eu-

roGIGA - ComPoSe IP04 - MICINN Project EUI-EURC-2011-4306.‡Research also supported by FCT grant

SFRH/BPD/66431/2009.§Email: [email protected]. Research also supported by FCT

grant SFRH/BPD/66572/2009.

edge are not taken into account for visibility purposes.Therefore, a point p is covered by an edge e if p is vis-ible from some interior point of e. As shown in Figure1, open edge guards can see considerably less polygonarea than the usual edge guards, and are therefore aninteresting topic of research on their own.

(a) (b)

uu v v

Figure 1: (a) The area covered by the open edge guarduv is shown in grey. (b) The area covered by the closededge guard uv is shown in grey.

Open edge guarding is a variation of the Art Galleryproblem that was first introduced by Viglietta [6] asa way to guard 3D polyhedra. This work was built onby Benbernou et al. [7] and Tóth et al. [5]. The latterstudied open edge guards and proved that there arepolygons that need at least bn

3 c guards to be covered,and that bn

2 c are always sufficient.This article is structured in the following way. Sec-

tion 1 introduces the concept of open edge guards andpresents results on the number of guards that cover or-thogonal and spiral polygons. Some results related tothe Fortress problem on simple and orthogonal poly-gons are also presented. The paper concludes withSection 2.

1 Open edge guards

Given a simple polygon P , GOE(P ) is the min-imum number of open edge guards that fullycover P and let GOE(n) := minGOE(P ) :P is a polygon of n vertices. Consequently, this sec-tion is devoted to calculate GOE(n) for differentclasses of polygons.

7

Results on open edge guarding of polygons

1.1 Orthogonal polygons

Bjorling-Sachs [1] proved that b 3n+416 c closed edge

guards are sufficient and sometimes necessary to fullycover an orthogonal polygon. This section shows thatbn

4 c open edge guards are sometimes necessary and al-ways sufficient to fully cover an orthogonal polygon.Given an orthogonal polygon P with n vertices, the

edges of P can be divided into four categories: north,south, west and east edges. North edges see the in-terior of the polygon below them, south edges see itabove them, east edges see it to their left and westedges see it to their right. Each of these four sets rep-resents a group of open edge guards that completelycovers P . In order to see this, choose any point p of P .From p, it is always possible to draw vertical segmentsthat will hit a north edge if it goes up from p and asouth edge if it goes down. The reasoning for the hor-izontal directions is similar. Therefore, the smallestof these four sets of edges proves the upper bound:any orthogonal polygon can be covered by bn

4 c openedge guards. Furthermore, there is an example of anorthogonal polygon that needs bn

4 c open edge guardsto be fully covered. Such polygon is very similar tothe one depicted in Figure 2, but where each spikeonly hides one point since there are no holes. Thesetwo bounds prove the following theorem.

Theorem 1 Any orthogonal polygon with n verticescan be covered by bn

4 c open edge guards, and in somecases this number is necessary.

Observe that this result essentially holds for orthog-onal polygons with holes, and the upper bound can beobtained in the same way it was explained above. InFigure 2 there is an example of a polygon with holesthat needs bn

4 c − 1 open edge guards to be fully cov-ered, since each marked point is seen by a differentopen edge guard.

Figure 2: A polygon with holes that needs bn4 c − 1

open edge guards to cover it, n = 44.

1.2 Spiral polygons

This section studies open edge guarding of spiral poly-gons, which will also be called spirals when it easesthe reading of the text. According to a previous work,bn+2

5 c closed edge guards are sufficient and sometimesnecessary to cover spiral polygons [8].

1.2.1 Tight bound on the number of openedge guards

In the example in Figure 3(a), each point marked onthe polygon needs a different open edge guard to coverit. Since this spiral has only one possible triangula-tion and there is one of these points per four trian-gles, this polygon needs dn−2

4 e open edge guards in or-der to be fully covered. Therefore, GOE(n) ≥ dn−2

4 e.This lower bound can be rewritten as bn+1

4 c, and it isproven below that this bound is tight.

(a) (b)

Figure 3: (a) No two of the marked points can becovered by the same open edge guard. (b) The twored open edge guards cover the whole spiral.

The boundary of each spiral can be decomposedinto a sequence of consecutive reflex vertices (calleda reflex chain) and a sequence of consecutive convexvertices (called a convex chain). The proof that anyspiral can be covered by bn+1

4 c open edge guards usesinduction on the number of edges of the polygon.The base case comprehends four cases. When the

spiral has four, five or six edges it is easy to see thatone open edge guard covers the whole polygon. Spi-rals with seven edges can be covered by two open edgeguards, since it suffices to place the guards on theedges of the convex chain that are intersected by theextensions of the first and last edges of the reflex chain(see Figure 3(b)).For the inductive step, suppose bn+1

4 c open edgeguards are sufficient to cover any spiral of n′ verticeswith n′ < n edges, n > 7. Let P be a spiral withn > 7 vertices, whose reflex chain is formed by thevertices r1, r2, . . . , rk. Now extend the edge r1r2

until it intersects some edge of the convex chain. Letv be the rightmost endpoint of the edge of the convexchain just intersected as shown in Figure 4(a). Theproof is now divided into four cases depending on thenumber of edges of the convex chain from c1 to v:(a) five or more than five edges, (b) four edges, (c)three edges and (d) two edges. For case (a), supposethe convex chain from c1 to v has at least five edges.Then draw the diagonal between r1 and c5, the fifthvertex of the convex chain. In this way, the spiralis partitioned into two spiral polygons: P ′ that hassix edges and so can be guarded with one open edgeguard and P ′′ with n−4 edges. For case (b), supposethe convex chain from c1 to v has four edges as shown

8


(c) (d)

c1

ct

v

P ′′P ′′

r3

r1

r2

c1

ct

P ′

r1

r2

v

P ′

(a) (b)

c1

r1

r2

ct

v

P ′ P ′′v

P ′′

c1

r1

ct

r2P ′

Figure 4: (a) The convex chain from c1 to v is formed by six edges. (b) The convex chain from c1 to v is formedby four edges. (c) The convex chain from c1 to v is formed by three edges: the first reflex vertex visible from vis r2; (d) the first reflex vertex visible from v is r3.

in Figure 4(b). Then draw the diagonal between r2

and c4, the fourth vertex of the convex chain. Thispartitions the spiral into two spiral polygons: P ′ thathas six edges and therefore can be guarded with oneopen edge guard and P ′′ with n− 4 edges.In case (c) the convex chain from c1 to v has three

edges and the situation is slightly different. As shownin Figures 4(c) and 4(d), draw the diagonal betweenv and the first visible reflex vertex starting from r2

(note that r2 can be such a vertex). This procedurepartitions the spiral into two polygons: P ′ that canbe guarded with one open edge guard and P ′′ with atmost n− 4 edges.Finally, case (d) in which the convex chain from

c1 to v has only two edges. In this case, draw thediagonal from v to the first visible reflex vertex afterr2. If there are no visible reflex vertices left, then thereflex chain is over and an open edge guard placed onthe second edge of the convex chain covers the wholespiral (see Figure 5(a)). If there is one visible reflexvertex then draw the diagonal as before, which willpartition the spiral into two polygons: P ′ that can beguarded with one open edge guard and P ′′ with atmost n− 4 edges (see Figure 5(b)).

(a) (b)ct

P ′′

ct

P ′′P ′

r1

c1

r2

v

c1

r1

r2

r3

vP ′

r4

Figure 5: The convex chain from c1 to v has two edges.(a) There is no reflex vertex visible from v besides r2.(b) The first reflex vertex visible from v is r4.

All the four cases described above end with a poly-gon P ′′ that has at most n − 4 edges, which meansthe inductive hypothesis can be applied. Therefore,P ′′ can be covered by b (n−4)+1

4 c = bn+14 c − 1 open

edge guards. Since polygon P ′ is covered exactly byone open edge guard, the whole spiral is covered by

bn+14 c open edge guards and this concludes the proof.

Theorem 2 Any spiral polygon with n vertices canbe covered by bn+1

4 c open edge guards, and in somecases this number is necessary.

1.2.2 Placing the minimum number of openedge guards

This section presents an algorithm to place the min-imum number of open edge guards that cover a spi-ral polygon P . The main idea of the algorithm is tobuild two sets simultaneously: G, which is the set ofopen edge guards, and H = h1, h2, . . . that is theset of points that guarantees G is of minimum size.The points that form set H are placed on the poly-gon in such a way that each open edge guard can seeonly one of them and is therefore associated with it.Consequently, |G| = |H|. Let r1, r2, . . . , rk be thereflex chain and c1, c2, . . . , cn−k the convex chain ofP . Moreover, let c1, c2, . . . , cn−k, rk, rk−1, . . . , r1 bethe sequence of n vertices of a spiral polygon P . Thesteps of the algorithm to place the minimum numberof open edge guards to cover P are depicted in Figure6 and explained in the following.

(a) (b)

h1

r1

c1

e1

e2

h2

r1

c1

e3

e4

e1

e2

Figure 6: (a) Finding an edge of the convex chain thatcovers point h1. (b) Finding an edge of the convexchain that covers point h2.

Set G is empty to start with. Let h1 ∈ P be a pointvery close to c1, which has to be covered. Draw theray −−→c1r1 that will intersect some edge of the convexchain that sees point h1. Let such edge be denotedby (e1, e2) and assign G ← (e1, e2). Secondly, find

9

Results on open edge guarding of polygons

the last reflex vertex rj that can be seen from e2 andconsider point h2, which is very close to rj along theedge (rj , rj+1). Then draw the ray −−−−→rjrj+1 that willintersect some edge of the convex chain that sees pointh2. Let such an edge be denoted by (e3, e4) and assignG ← G ∪ (e3, e4). Repeat this last step until allreflex vertices and cn−k are guarded.This algorithm selects the edges ejej+1 of the con-

vex chain that will be part of set G, which fully coversany spiral polygon since it totally covers its convexchain.

Lemma 3 The algorithm described in this sectionbuilds a set H of points interior to P such thatGOE(P ) ≥ |H|.

Theorem 4 The algorithm described in this sectionplaces the minimum number of open edge guardsneeded to cover a spiral polygon in O(n) time.

Proof. Let G be the set of open edge guards builtby the algorithm to cover spiral polygon P and Hthe set of points of P in which each point is cov-ered by a different guard. That is, the set of pointsthat are placed in such way that the According toLemma 3, GOE(P ) ≥ |H| but since |H| = |G| thenGOE(P ) ≥ |G| and therefore G is a minimum set ofopen edge guards. Regarding the time complexity,each edge of the convex chain is only processed oncewhilst analysing the rays −−−−→rjrj+1. In the same way,each edge of the reflex chain is checked once to findthe last reflex vertex that is visible from the chosenedges. Consequently, each vertex of the spiral polygonis analysed just once by the algorithm and thereforeit runs in linear time.

1.3 Fortress problemThis section is devoted to another variation of theArt Gallery problem called the Fortress Problem. In-stead of guarding the interior of a simple polygon, theFortress Problem variation focuses on monitoring theexterior of a polygon. Choi et al. [2] proved that theexterior of any simple polygon can be covered by dn

3 eclosed edge guards and that these guards are neces-sary to cover the exterior of convex polygons. In thecase of open edge guards, this problem is trivial sinceit is easy to see that every edge will be needed as aguard to cover the exterior of a convex polygon.The natural following step is to study orthogonal

polygons. Again, Choi et al. [2] proved that the ex-terior of any orthogonal polygon can be covered bybn

4 c+1 edge guards and that this number can be nec-essary. The proof of the following theorem is omitted,but it is based on the technique of dividing the edgesaccording to their orientation, as introduced in Sec-tion 1.1. The lower bound is given by the orthoconvexpolygon depicted in Figure 7.

Theorem 5 The exterior of any orthogonal polygonwith n vertices can be covered by n

2 + 2 open edgeguards, and in some cases this number is necessary.

Figure 7: Orthoconvex polygon that needs n2 +2 open

edge guards to be covered.

2 Final remarksWe studied other classes of polygons, such as mono-tone polygons, as well as other geometric configura-tions. There are monotone polygons that need bn

3 copen edge guards in order to be fully covered and webelieve this bound is tight. Furthermore, this boundis proved for open mobile guards, which can patroledges and diagonals of a polygon. This type of cover-age has also been studied for other polygons.

References[1] I. Bjorling-Sachs. Edge guards in rectilinear polygons.

Comput. Geom. Theory Appl., 11(2):111–123, 1998.

[2] A. Choi and S. Yiu. Edge guards for the fortress prob-lem. Journal of Geometry, 72:47–64, 2001.

[3] D. Lee and A. Lin. Computational complexity of artgallery problems. IEEE Transactions on InformationTheory, 32(2):276–282, 1986.

[4] T. Shermer. Recent results in art galleries. Proceedingsof the IEEE, 80(9):1384–1399, 1992.

[5] C. D. Tóth, G. T. Toussaint, and A. Winslow. Openguard edges and edge guards in simple polygons. InA. Márquez, P. Ramos, and J. Urrutia, editors, Com-putational Geometry, volume 7579 of Lecture Notes inComputer Science, pages 54–64. Springer Berlin Hei-delberg, 2012.

[6] G. Viglietta. Guarding and Searching Polyhedra. PhDthesis, University of Pisa, 2012.

[7] G. Viglietta, N. Benbernou, E. D. Demaine, M. L. De-maine, A. Kurdia, J. O’Rourke, G. T. Toussaint, andJ. Urrutia. Edge-guarding orthogonal polyhedra. In23rd Canadian Conference on Computational Geome-try, 2011.

[8] S. Viswanathan. Tight bounds for the number ofedge guards for spiral polygons. Journal of Geome-try, 51:178–186, 1994.

10



Ana Paula Tomás∗

DCC & CMUP, Faculdade de CiênciasUniversidade do Porto, Portugal

Abstract

An orthogonal polygon of P is called thin if the dualgraph of the partition obtained by extending all edgesof P towards its interior until they hit the boundaryis a tree. We show that the problem of computing aminimum guard set for the vertices of a thin orthogo-nal polygon is NP-hard either for guards lying on theboundary, or on vertices or anywhere in the polygon.

Introduction

The classical art gallery problem for a polygon P asksfor a minimum set of points G in P such that everypoint in P is seen by at least one point in G (the guardset). Many variations of art gallery problems havebeen studied over the years to deal with various typesof constraints on guards and dierent notions of visi-bility. In the general visibility model, two points p andq in a polygon P see each other if the line segment pqcontains no points of the exterior of P . The set V (v)of all points of P visible to a point v ∈ P is the visibil-ity region of v. A guard set G for a set S ⊆ P is a setof points of P such that S ⊆ ∪v∈GV (v). Two pointsvi and vj are equivalent for the visibility relationif V (vi) ∩ S = V (vj) ∩ S. If V (vj) ∩ S ⊂ V (vi) ∩ Sthen vi strictly dominates vj , and vi can replace vj

in an optimal guard set of S. Guards that may lieanywhere inside P are called point guards whereas ver-tex guards are restricted to lie on vertices and bound-

ary guards on the boundary. Combinatorial upperand lower bounds on the number of necessary guardsare known for specic settings (for surveys, refer toe.g. [8, 10]). The fact that some art gallery problemsare NP-hard [5, 9] motivates the design of heuristicand metaheuristic methods for nding approximatesolutions and also the study of more specic classesof polygons where some guarding problems may betractable [1, 2, 3, 6]. In this paper, we address theproblem of guarding the vertices of orthogonal poly-gons, which is known to be NP-hard for generic or-

∗Email: [email protected]. Research partially supportedby the European Regional Development Fund through theprogramme COMPETE and by the Portuguese Governmentthrough the FCT Fundação para a Ciência e Tecnologia un-der the project PEst-C/MAT/UI0144/2011.

thogonal polygons [4]. We show that the problem isNP-hard also for the family of thin orthogonal poly-

gons, which consists of the orthogonal polygons suchthat the dual graph of the corresponding grid parti-

tion ΠHV (P ) is a tree. ΠHV (P ) is obtained by addingall horizontal and vertical cuts incident to the reexvertices of P (see Fig. 1). Our proof is inspired in [4]

Figure 1: Orthogonal polygons, grid partitions anddual graphs: (a) ΠHV (P ) and its dual graph in gen-eral; (b) a thin orthogonal polygon; (c) a thin orthog-onal polygon that is a path orthogonal polygon.

although the need to obtain thin orthogonal polygonsled to novel aspects in the construction. The classof thin orthogonal polygons contains the class of thinpolyomino trees introduced in [1], for which the au-thors conjecture that the guarding problem under thegeneral visibility model has a polynomial-time (exact)algorithm. To the best of our knowledge, this prob-lem is open. In [12], we give a linear-time algorithmfor computing an optimal vertex guard set for anygiven path orthogonal polygon (for which the dualgraph of ΠHV (P ) is a path graph), and prove tightlower and upper bounds of dn/6e and bn/4c for theoptimal solution for the subclass where all horizontaland vertical cuts intersect the boundary at Steinerpoints. Since the thin grid orthogonal polygons be-long to this class, our work extends results previouslyknown for the spiral thin grid orthogonal polygonsand the MinArea grid orthogonal polygons [6] (forwhich the minimum vertex guard sets have exactlybn/4c and dn/6e guards) and somehow explains whythe MinArea grid orthogonal polygons were consid-ered representative of extremal behaviour [11]. Theresult that supports our proof allows us to concludethat a minimum guard set for the vertices of a pathorthogonal polygon can be found in linear-time.

11


In rest of the paper, we show that computing aminimum guard set for the vertices of a thin orthogo-nal polygon (GVTP) is NP-hard, either for boundaryguards, vertex guards or point guards.

1 Hardness for boundary guards

Theorem 1 GVTP for thin orthogonal polygons is

NP-hard for boundary guards.

For the proof, we dene a reduction from thevertex-cover problem in graphs (VC) to GVTP withboundary-guards. VC, known to be NP-complete, isthe problem of deciding whether a graph G = (V,E)has a vertex-cover S of size |S| ≤ k, for k integer. Avertex-cover of G is a subset S ⊆ V such that for eachedge (u, v) ∈ E, either u ∈ S, or v ∈ S, or both.The thin orthogonal polygon we construct for a

given graph G = (V,E) is a large square with |E|tiny d-gadgets attached to its bottom. In Fig. 2 wesketch this construction and in Fig. 3 we present thedouble gadget (d-gadget) dened for the proof.

Figure 2: From VC to GVTP with boundary guards:the representation of G = (u, v, w, (u, v), (u,w));the edges of G are mapped to the points uv and uw(that will be replaced by tiny d-gadgets) and the ver-tices are mapped to the segments u, v and w.

We dene the side-length of this square to be L∆,with L = 1 + 2|V | + 3|E| and ∆ = 10L. Consider-ing V = v1, v2, . . . , vn sorted, we denote by E+

i thesubset of all edges (vi, vj) ∈ E such that i < j, alsosorted by increasing value of j. In the construction wefollow these orderings: for each i, we represent vi by asegment of length ∆ on the top edge of the square andthe edges in E+

i as middle points of |E+i | consecutive

segments of length 2∆ on the bottom edge, placedbetween the projections of vi and vi+1, and with sep-aration gaps of length ∆ between each other. Thesquare is implicitly divided into L slabs of length ∆,and we leave the rst slab empty and an empty slabbetween consecutive items.

Figure 3: A sketch of the d-gadget Ξij dened for theedge (vi, vj). The vertices on the left side are labelledfrom N1 to N18 in CW order and on the right sideare labelled fromM1 toM18 in CCW order. B ≡ N11

and A ≡M11 are the two distinguished vertices.

The d-gadget associated to the edge (vi, vj) ∈ E+i ,

denoted by Ξij , is dened as follows. Let Oij be thepoint that represents the edge (vi, vj) and AiBi andAjBj the segments associated to vi and vj . Togetherwith Oij , these segments dene two visibility coneswith apex Oij . By a slight perturbation, we candecouple the two cones, and move the new apexesto the distinguished vertices (B and A) of a tiny d-gadget Ξij . The structure of this gadget will x seg-ment AiBi (resp. AjBj) as the portion of the bound-ary of the polygon that A (resp. B) sees above line X(i.e, above the gadget). We use VC instead of theminimum 2-interval piercing problem used in [4] inorder to be able to control the aperture of visibilitycones and also the structure of the thin orthogonalpolygon obtained in the reduction. Some of the ver-tices of a d-gadget can only be guarded by a local

guard (i.e., a guard below line X), for instance, thevertices M16, M12, M8, M7 and M5 on its right partand N16, N12, N8, N7 and N5 on the left part. Forevery d-gadget, at least three local boundary-guardswill be needed to guard these vertices and no threesuch guards can see both A and B if they see all thesevertices. Moreover, one can always locate three localboundary-guards that see all the gadget vertices otherthan A (namely, at N8, N1 and M8) or other than B(namely, at N8, M1 and M8). Another guard is re-quired to guard the unguarded vertex but it does notneed to be local. As we will see, this guard can belocated on the portion of the top edge of the polygonseen from the unguarded vertex.

We dene the coordinates of the vertices of Ξij

w.r.t. a cartesian system ROijwith origin at Oij . By

construction, the x-coordinates of the points Ai, Bi

and Oij w.r.t. a cartesian system xed at the bottomleft corner of the large square are given by

x′Ai= (2i− 1 + 3

∑k<i |E

+k |)∆

x′Bi= x′Ai

+ ∆x′Oij

= x′Bi+ 2∆ + 3∆ |E+

i ∩ (vi, vj′) : j′ < j|

12


and, consequently, if we dene xi and xj as xi =(x′Oij

− x′Bi)/∆ and xj = (x′Aj

− x′Oij)/∆, then,

w.r.t. the cartesian system ROij, we have

Ai = (−(xi + 1)∆, L∆) Aj = (xj∆, L∆)Bi = (−xi∆, L∆) Bj = ((xj + 1)∆, L∆)

for integers xi ≥ 2 and xj ≥ 2. Then, we dene A andB as the intersection points of the supporting lines of−−−→OijAi and

−−−→OijBj with the line y = −4L, that is, as

A = (4xi + 4,−4L) and B = (−4xj − 4,−4L). So,

the rays−−→AAi and

−−→BBj share the supporting lines of

the initial rays−−−→OijAi and

−−−→OijBj . The aperture of

the visibility cone CA = cone(A,AiBi) is determinedby the vertices M1 and M13. We selected M1 as the

intersection of−−→AAi with the line y = −2L and M13

as the intersection of−−→ABi with the line y = −3L.

Therefore, M1 = (2xi +2,−2L) andM13 = (τi,−3L),with τi = 3xi + 3 + ∆

∆+4 , because the straight linesAAi and ABi are given by the following equations.

AAi : y =−Lxi + 1

x

ABi : y =−L(∆ + 4)xi(∆ + 4) + 4

x+4L∆

xi(∆ + 4) + 4

Similarly, the vertices N1 and N13 determine theaperture of the visibility cone CB = cone(B,AjBj),being N1 = (−2xj − 2,−2L) the intersection of

−−→BBj

with y = −2L and N13 = (τj ,−3L) the intersection

of−−→BAj with y = −3L, with τj = −3xj − 3 − ∆

∆+4 .The coordinates of the vertices of Ξij are

M1 = (2xi + 2,−2L) M2 = (2xi + 2,−4L)M3 = (2xi + 1,−4L) M4 = (2xi + 1,−3L)M5 = (2xi,−3L) M6 = (2xi,−6L)M7 = (2xi + 1,−6L) M8 = (2xi + 1,−5L)M9 = (τi,−5L) M10 = (τi,−4L)A = (4xi + 4,−4L) M12 = (4xi + 4,−3L)M13 = (τi,−3L) M14 = (τi,−2L)M15 = (7L,−2L) M16 = (7L,−L)M17 = (2xi + 2,−L) M18 = (2xi + 2, 0)

with the Nk = (−αxj−β, γ) iMk = (αxi +β, γ), for1 ≤ k ≤ 18. Therefore, the coordinates of the verticescan be dened by rational numbers represented bypairs of integers bounded by a quadratic polynomialfunction on the size of the graph.We can prove that: the dual graph of the grid par-

tition of the resulting polygon is a tree; M16, M12,M8, M7, M5, and N16, N12, N8, N7 and N5 requirelocal guards; the boundary of Ξij imposes no restric-tion on the propagation of the corresponding visibilitycones CA and CB ; the unique point on the boundaryof Ξij that sees both A and N16 is M1 (similar for B,M16 and N1); the three local guards N8, N1 and M8

jointly see all the gadget vertices other than A (simi-lar for N8, M1 and M8 and B). Lemma 2 states thenal result we need to conclude the proof and can beshown as Lemma 2.2. of [4].

Lemma 2 The thin orthogonal polygon P that is ob-

tained can be guarded by 3|E|+ k boundary guards if

and only if the there is a vertex-cover of size k for the

instance graph G = (V,E).

2 Hardness for vertex guards

Theorem 3 GVTP is NP-hard for thin orthogonal

polygons with vertex guards.

For the proof, we can adapt the previous construc-tion, following the idea of [4], as sketched in Fig. 4.

Figure 4: The reduction from VC to GVTP withvertex guards for G = (u, v, w, (u, v), (u,w)). Eargadgets are attached to the right endpoints of the seg-ments. Each ear gadget requires a local guard on avertex of the shaded region (to guard Z2).

We consider the polygon obtained previously andattach a tiny ear gadget to the right endpoint of eachline segment AiBi, for each vi ∈ V . The local verticesof the ear gadget attached to Bj , w.r.t. the cartesiansystem xed at the bottom left corner of the largesquare, can be dened as

Z1 = ((x′j + 1)∆, L(∆ + 1) + 1)Z3 = ((x′j + 1)∆ + 1, L(∆ + 1))Z2 = ((x′j + 1)∆ + L, L(∆ + 1))

and ((x′j + 1)∆ + L, L(∆ + 1) + 1). The separationslabs guarantee that the dual graph of ΠHV (P ) forthe new polygon P is still a tree, as required. The eargadgets are dened in such a way that the vertex A ofΞij cannot see any vertex of an ear-gadget except forBi. Otherwise, A would see points on the boundaryof P arbitrarily closed to Bi but to the right of Bi,which is impossible by the denition of the visibilitycone CA. The height of the ear gadget prevents B from

13


seeing any local vertex of the ear-gadget attached toAjBj . For each j ≥ 2, it is sucient to guarantee

that, for all Ξij , the intersection point of the ray−−→BBj

with the vertical edge incident to the vertex labeledZ3 is below Z3. This holds for all i if it holds for B in

the rightmost d-gadget Ξi′j , since the rays−−→BBj are

sorted by slope around Bj .Each ear-gadget needs a local guard that must be

located in one of the vertices of the shaded regionand none of these vertices sees a local vertex of a d-gadget. This means that these guards cannot replaceany guard located in a segment. Since any guard lo-cated on a segment can move to the segment rightendpoint to become a vertex-guard, without loss ofvisibility, we can adapt the proof of Lemma 2 to showLemma 4.

Lemma 4 The thin orthogonal polygon P that is ob-

tained can be guarded by |V |+3|E|+k boundary guards

if and only if the there is a vertex-cover of size k for

the instance graph G = (V,E).

3 Hardness for point guards

Theorem 5 GVTP is NP-hard for thin orthogonal

polygons with point guards.

For the proof, we construct a reduction from theminimum line cover problem (MLCP), as in [4].MLCP is NP-hard [7]. Given a set L = l1, . . . , lnof lines in the plane, MLCP is the problem of ndinga set of points of minimum cardinality such that eachline l ∈ L contains at least one point in that set.Without loss of generality, we consider that L con-

tains neither vertical nor horizontal lines. The poly-gon constructed for the reduction is obtained by at-taching single-gadgets (called s-gadgets) to a boundingbox B(L) that contains all intersection points of pairsof lines in L in its interior. The idea of this construc-tion is sketched in Fig. 5.

Figure 5: The reduction fromMLCP to GVTP withguards anywhere. Each tiny box on the bottom rep-resents an s-gadget (note that not all lines intersectedthe bottom edge of the dashed bounding box). On theright, an s-gadget in detail (the vertices are labelledfrom M1 to M14, in CCW order, A ≡M11).

In order to guarantee that a thin orthogonal poly-gon is obtained, we dene a new type of s-gadget,sketched on Fig. 5, where M1 and M13 reduce thevisibility cone CA to the line LA. Moreover, we hadto restrict the locations of s-gadgets to the bottomedge of B(L), in contrast to [4]. This can be done be-cause, for a suciently large bounding box, all lineswill intersect the bottom edge of B(L), as there are nohorizontal lines in L. At least a local guard is neededfor each s-gadget. As for the d-gadgets, taking intoaccount the relative positions of intersections of thelines with the bottom line (i.e., of vertices M1), andtheir slopes, we can dene the vertices of the tiny s-gadget in such a way that M8 sees M12 and M7, andall local vertices except for A. We can conclude thatthe vertices of P can be guarded by n + k guards ifand only if there is a cover for L of size k.

References

[1] T. Biedl, M. T. Irfan, J. Iwerks, J. Kim and J. S. Mit-chell, Guarding polyominoes, in: Proc. SoCG 2011,ACM, 2011, 387396.

[2] A. Bottino and A. Laurentini, A nearly optimal al-gorithm for covering the interior of an art gallery,Pattern Recognition 44 (2011), 10481056.

[3] M. C. Couto, P. J. de Rezende, and C. C. de Souza,An exact algorithm for minimizing vertex guards onart galleries, Int. T. Oper. Res. 18 (2011), 425 448.

[4] M. J. Katz and G. S. Roisman, On guarding the ver-tices of rectilinear domains, Computational Geometry Theory and Applications 39 (2008), 219228.

[5] D. T. Lee and A. K. Lin, Computational complexityof art gallery problems, IEEE Transactions on Infor-mation Theory 32 (1986), 276282.

[6] A. M. Martins and A. Bajuelos, Vertex guards ina subclass of orthogonal polygons, Int. J. ComputerScience and Network Security, 6 (2006), 102108.

[7] N. Megiddo and A. Tamir, On the complexity oflocating facilities in the plane, Oper. Res. Lett. 1(1982), 194197.

[8] J. O'Rourke, Art Gallery Theorems and Algorithms,Oxford University Press, 1987.

[9] D. Schuchardt and H. Hecker, Two NP-hard problemsfor ortho-polygons, Math. Logiv. Quart. 41 (1995),261267.

[10] J. Urrutia, Art gallery and illumination problems, in:J.-R. Sack and J. Urrutia (eds), Handbook on Com-putational Geometry. Elsevier, 2000.

[11] A. P. Tomás, A. Bajuelos and F. Marques, On visi-bility problems in the plane solving minimum ver-tex guard problems by sucessive approximations, in:Proc. Int. Symp. Articial Intelligence and Mathe-matics (ISAIM), Florida, 2006.

[12] A. P. Tomás. Guarding path orthogonal polygons,DCC&CMUP, Univ. Porto, 2013 (in prep).

14


Solving common inuence region queries with the GPU

Marta Fort and J.Antoni Sellarès∗

Departament Informàtica, Matemàtica Aplicada i Estadística. Universitat de Girona.

Abstract

In this paper we propose and solve common inu-

ence region queries. We present GPU parallel algo-rithms, designed under CUDA architecture, for ap-proximately solving the studied queries. We also pro-vide and discuss experimental results showing the ef-ciency of our approach.

Introduction

Common inuence region queries are related to thecapacity of two sets of facilities of dierent type, com-petitive and collaborative, of attracting customers.Solutions to common inuence region queries help de-cision makers to develop competitive-collaborative op-portunities.

Papers [2, 3, 1] deal with a related problemwhich tries to maximize the area of the Voronoiregion of a new non-weighted facility. Fort andSellarès [4] present an algorithm for optimizing k-nearest/farthest inuence regions of a set of non-weighted facilities.

The advancement in GPUs (Graphics ProcessingUnits) hardware design, together with CUDA (Com-pute Unied Device Architecture), make them attrac-tive to solve problems which can be treated in parallelas an alternative to CPUs. General-Purpose comput-ing on GPUs (GPGPU) is playing an increasing rolein scientic computing applications, which range fromnumeric computing operations to data mining or geo-metric processing, where the potential of the GPU fordelivering real performance gains on computationallycomplex, large problems is demonstrated.

By working towards practical solutions, since ex-act algorithms to solve the common inuence regionqueries are hard to implement and quite slow in prac-tice, we explore a GPU parallel approach, designedunder CUDA architecture, for solving the queries ap-proximately. We also provide and discuss experimen-tal results obtained with the implementation of ouralgorithms that show the eciency and scalability ofour approach.

∗Email:(mfort,sellares)@imae.udg.edu.Authors research supported by TIN2010-20590-C02-02.

1 Preliminaries

1.1 The weighted area of a region

Let R = R1, · · · , Rm be a partition of the domainD. We associate with each region Ri a non-negativenumber wi, called the weight of Ri.

We dene the weighted area of the region R ⊂ D,denoted w(R), as:

w(R) =∑

j=1,··· ,mwj µ(R ∩Rj) ,

where µ(R ∩ Rj) denotes the area of the subregionR ∩Rj .

1.2 CUDA architecture

CUDA is a parallel computing architecture thatmakes GPUs accessible for computation like CPUs.The CUDA processors, which can be executed in par-allel, are referred as threads, and each thread exe-cutes the instructions contained in the so called ker-nels in parallel. Each thread computation is indepen-dent from the others, however, there exist some read-modify-write atomic operations called atomic func-tions. They read and return the value stored in amemory position, operate on it and store the resultwithout allowing, during the whole process, any otheraccess to that memory position. These operations al-low the users to obtain global results when severalthreads access to the same memory position. For in-stance we can obtain a global sum, maximum or min-imum in a specic position by using them.Several types of memories can be used, data stored

in global memory are accessible by every thread andare visible from the CPU, global memory is wheremore data can be allocated, but is the slowest accesstime memory. Shared memory and registers are thefastest memory, shared memory can be accessed by allthe threads of the same block, and registers store thelocal variables of the threads. The number of accessesto memory are reected in the execution times of thealgorithms. Thus they are provided in the complexityanalysis, ν read or written values to global memoryare represented by rgν and wg

ν , and to shared memorythey are denoted by rsν and ws

ν .

15

Common inuence region queries

2 Inuence regions

Let S be a set of n points included in a boundeddomain D of the Euclidean plane. Each point s ∈ Sis associated with a positive real weight ws > 0. Theweighted distance ds(q) from an arbitrary point q ∈ Dto the point s ∈ S is dened as ds(q) = (1/ws) d(s, q),where d(p, q) denotes the Euclidean distance betweenpoints p, q.The k-inuence region of a point s ∈ S, denoted

Ik(s, S), is the set of points of D having s amongtheir k-nearest points in S. For any point q ∈ D,denote by nk(q, S) the weighted distance from q to itsk-th nearest point in S, i.e. the weighted distance tothe point of S that ranks number k in the orderingof the points by increasing weighted distance from q.We have:

Ik(s, S) = q ∈ D | ds(q) ≤ nk(q, S) .

From the denition follows that the 1-inuence regionof a site s is its 1-Voronoi region, and that Ik(s, S) ⊂Ik+1(s, S).A k-inuence region is bounded by bisectors of

points in S. In general, k-inuence regions need not tobe convex, nor simply connected, nor even connected.Two k-inuence regions may share disconnected edges(see Figures 1 a) and b)).

2.1 Common inuence regions

Let P andQ be nite disjoint sets of n andm weightedpoints, respectively, within the bounded domain D ofthe Euclidean plane.The (k, k′)-Common Inuence Region of p ∈ P , q ∈

Q, denoted Ck,k′(p, q, P,Q), is the set of points of Dhaving p among their k-nearest points in P and at thesame time having q among their k′-nearest points inQ:

Ck,k′(p, q, P,Q) = Ik(p, P ) ∩ Ik′(q,Q) .

In Figure 1 c) we can see an example of a (5, 2)-common inuence region painted in green.

a) b) c)

Figure 1: a) 5-inuence region, b) 2-inuence region, c)

(5, 2)-common inuence region.

2.2 Common inuence region queries

Let P andQ be nite disjoint sets of n andm weightedpoints, respectively, within the bounded domain D of

the Euclidean plane, and R be a weighted partitionof D. We study the following queries:

Feasible pairs query. Given xed non-empty setsP ′ ⊆ P and Q′ ⊆ Q and a real positive number W0,nd all pairs (p, q), p ∈ P ′ and q ∈ Q′, such that theweighted area w(Ck,k′(p, q, P,Q)) ≥ W0.

Feasible partners query. Given xed non-emptysets P ′ ⊆ P and Q′ ⊆ Q and a real positive numberW0, nd all points p ∈ P ′, such that for each q ∈ Q′

the weighted area w(Ck,k′(p, q, P,Q)) ≥ W0.

The parameters k, k′, W0, I0 should be chosen byan expert, according to the localization of p with re-spect the points ofQ′ and the cardinality ofQ′, duringan iterative what-if analysis process.

To obtain w(Ck,k′(p, q, P,Q)) = w(Ik(p, P ) ∩

Ik′(q,Q)), we can rst compute the overlay intersec-

tion between Ik(p, P ), Ik′(q,Q) and the regions of theweighted partition R. Since each maximal connectedregion of the resulting intersection is contained in aregion of R, it has univocally associated a weight.Consequently, the weighted area w(Ck,k′(p, q, P,Q))can be written as a sum of functions that depends onthe location of p and q. Since computing the maximalconnected regions together with their weighted areais dicult, it is also dicult computing the weightedarea of a common inuence region and solving com-mon inuence region queries. This has motivated usto explore an alternative GPU parallel approach, de-signed under CUDA architecture, for approximatelysolving common inuence region queries.

3 Solving common inuence re-

gion queries using CUDA

The weighted areas of the common inuence regionsare estimated by using a discretization of the domainD. An axis-parallel rectangular grid of size H × Wis superimposed on the domain D dening a set S ofr points corresponding to the geometric center of thegrid cells which are weighted according to the domainpartition P. As it is obvious, the bigger the size ofthe used grid the more accurate the obtained commoninuence region. The weighted area is estimated bysumming up the weights of all the points of S con-tained in the common inuence region. Notice that,depending on the domain weighted partition, smallchanges in the common inuence region can producebig changes in its weighted area.

Solving the dened queries requires an initial stepconsisting in computing the k-nearest neighbor dis-tance from each point of S to the points in P and thek′ nearest neighbor to the points in Q. Notice thateven though the queries subsets P ′ and Q′ are used,

16


the k and k′-neighbor distances are always referred toP and Q.

3.1 k-nearest neighbor distance com-

putation

We start by computing the weighted distance of eachpoint s ∈ S to its k-th nearest neighbor in P , consid-ering all the points in parallel. With this aim, we haveextended the CudaKNN algorithm described by Lianget al. [5] to handle the weighted case. The algorithmcomputes for each p ∈ P the weighted distances toall the points in S using shared memory and makingthe threads in a block cooperate to determine, afterexploring all the point in P , the k-nearest neighborsof s. We transfer the points of S and P from the CPUto the GPU, then we compute the k-weighted nearestdistance of each point in S to P and store all them inglobal memory in a 1D array nPk

of size r.In the same way, we also transfer the points of S

and Q from the CPU to the GPU, compute the k′-weighted nearest distance of each point in S to Q andstore them in a 1D array nQk′ of size r.

3.2 Inuence region queries resolution

Given P ′ ⊆ P and Q′ ⊆ Q of size n′ and m′ respec-tively, and assuming that the k and k′-nearest neigh-bor distances from each point of S to P and Q havebeen computed, we explain how the proposed queriescan be solved.Our approach for solving the feasible pairs and the

possible partners queries follow a very similar proce-dure.

Feasible pairs query

The solution of the feasible pair query is reportedgiving the number of feasible pairs and the list con-taining them. In order to solve it we consider n′m′

threads, an integer to store the number of feasiblepairs and an 1D-array of size n′m′ to store the fea-sible pairs. Each thread estimates the weighted areawp,q = w(Ck,k′(p, q, P,Q)) of one pair (p, q). In thecase that wp,q gets the minimum required value W0,the number of feasible pairs is incremented by one,and the pair is stored in the rst empty position ofthe feasible pairs array.The weighted area wp,q is estimated by considering

all the points s ∈ S, computing the distances dp(s)and dq(s) and comparing them with nk(s, P ) andnk′(s,Q) which are the ones stored in nPk

and nQk′ .Thus s ∈ Ck,k′(p, q, P,Q) whenever dp(s) ≤ nPk

(s)and dq(s) ≤ nQk′ (s). In such a scenario, where allthreads check all the points in S, shared memoryshould be used. The B threads in a block cooperateloading the space points and their neighbor distancesfrom global to shared memory. This requires usingthree shared memory arrays of size B per block: Ss

which stores B weighted space points, and nsPk, and

nsQk′ storing their corresponding k and k′-neighbor

distances to P and Q, respectively. The i-th thread ofthe block reads from global memory the point si ∈ Sand its corresponding k and k′ nearest neighbor dis-tances, and stores them in the i-th position of Ss andnsPk

and nsQk′ , respectively. When all the threads

in the block have nished loading this information,the rst B points of S can be analyzed. Then eachthread determines which of these points are containedin its corresponding common region Ck,k′(p, q, P,Q)and accordingly accumulates their weights in a regis-ter. Once all the threads of the block have checkedall the already loaded points, the next B points of Sand their distances are loaded to shared memory andanalyzed. We keep on proceeding similarly until allthe points in S have been handled, and consequentlywpq has been estimated.

In order to obtain correct results it is importantthat the threads in a block wait each other in somecritical points. In fact, two synchronization pointsare needed, one after transferring the correspondingweighted point of S with its two associated distancesand the other when all the already stored points havebeen checked, just before starting transferring newinformation to shared memory. Otherwise there isno guarantee that all the threads in the block havenished with their tasks. The implementation has tobe carefully done when r, the number of points in S, isnot a multiple of B, the number of threads in a block,to avoid accesses to non-existent memory positions,but guaranteing that all the points of S are loaded toshared memory.

Feasible partners query

To solve this query we use an array fp, of size n′,which is used as a boolean array. At the end of theprocess, positions containing a 1 correspond to thepoints of P ′ having all the points of Q′ as feasiblepartners.

After initializing the array fp with ones, we proceedas before estimating the weighted common inuenceareas of all the m′n′ pairs in P ′ ×Q′. When a threadnishes estimating its corresponding weight wpq, itchecks wether wpq < W0, and in such a case, it setsfp[p] to 0. Thus, only for those points p achieving theminimum inuence value for all the points q ∈ Q′ wewill have fp[p] = 1 at the end of the process.

The number, n′f , and the list, f , of feasible partners

can be obtained by performing a prex sum on thearray fp, which is stored in ps. After allocating anarray of size n′

f , n′ threads are considered, each thread

checks wether its point of P ′ is a feasible partner ofQ′, by looking at its corresponding position of fp. Ifit is so, the point of P ′ is stored in f in the positiongiven by ps.

17

Common inuence region queries

3.3 Complexity analysis

Computing simultaneously for the r points of S the k-neighbor distances with respect to the set of n pointsby using the CudaKNN algorithm [5] requires O(rnB)total work, where B is the number of threads perblock. The computation is done in three dierentsteps, that have O(r), O(B) and O(k + n/B) oper-ations per thread, respectively. Thus, the initial stepin our case requires O(r(n+m)B) total work.

Solving the feasible pairs query requires n′m′

threads performing O(r) operations each, leading toO(rn′m′) total work. Concerning memory accesses, inthe worst case O(rgn′m′r/B+ws

n′m′r/B+rsn′m′r+wgn′m′)

accesses are required. Finally, the global memoryneeded is 3r + n′ + m′ + n′m′ + 1, while 3B oatsper block are required in shared memory.

The dierences between solving the feasible pairsquery and the feasible partners one, before ndingthe list of feasible partners, do not change the workper thread neither the global work nor the memoryaccesses. However, the global memory requirementsare reduced to 3r + 2n′ +m′. Finding the list f onlyincreases the global memory requirements in n′ + n′

f .

4 Experimental Results

In this section, we present several experimental resultsobtained with the implementation of our algorithms.In all the experiments the points of P and Q are ran-domly distributed in the domain and their weights arerandomly obtained integers between 1 and 15. Theyare based on the weighted domain partition presentedin Figure 2. The points of the domain are paintedin a purple color gradation, the darker the color thesmaller the density value.

Figure 2: Domain weighted partition according to the

population density.

In Figure 3, we provide the solution of a feasiblepartners query. Figure 3 a) shows the sets P , Q, P ′

and Q′, that have 100, 100, 25 and 10 points, respec-tively. Points of sets P ′ and Q′ are represented bygrey squares and green triangles, and the points of Pand Q not in P ′ and Q′, by red squares and blue trian-gles, respectively. The orange squares of Figure 3 b)are the points of P ′ conforming the solution of thefeasible partners query dened by P , Q, P ′, Q′ for

k = k′ = 15, when the minimum required inuencevalue is 20.

a)

b)

Figure 3: a) Points of the sets P , Q, P ′, Q′. b) Solutions

of the feasible partners query.

For a grid of size 400 × 400, n = 500, n′ = 50,m = 200, m′ = 25, k = 20 and k′ = 20, the runningtimes needed to solve a feasible pairs query and a fea-sible partners query are 12 and 16 milliseconds. Us-ing the CPU sequential version of the algorithms thetimes become 347 and 351 miliseconds, respectively.In the CPU version, the feasible partners query canbe solved in 62 miliseconds if point p ∈ P ′ is set asa not feasible partner at the rst q ∈ Q′ for whichw(Ck,k′(p, q, P,Q)) < W0. This can not be done inthe GPU, because threads cooperate transferring in-formation to shared memory.

References

[1] O. Cheong, A. Efrat, and S. Har-Peled, Finding a guard

that sees most and a shop that sells most, Discrete Com-put. Geom. 37(4) (2007), 545563.

[2] M. Denny, Solving geometric optimization problems us-

ing graphics hardware, Comput. Graph. Forum 22 (2003),no. 3, 441452.

[3] F. K. H. A. Dehne, R. Klein, and R. Seidel, Maximizing a

Voronoi region: the convex case, Int. J. Comput. Geome-try Appl. 15(5) (2005), 463476.

[4] M. Fort and J.A. Sellarès, GPU-Based Inuence Regions

Optimization, ICCSA (2012), 253266.

[5] S. Liang, Y. Liu, C. Wang, L. Jian, A CUDA-based parallel

implementation of k-nearest neighbor algorithm, IEEE'09(2009), 291296.

18


Reporting ock patterns on the GPU

Marta Fort, J.Antoni Sellarès, and Nacho Valladares∗

Departament Informàtica, Matemàtica Aplicada i Estadística. Universitat de Girona.

Abstract

In this paper we study the problem of nding ockpatterns in a set of trajectories of moving entities. Aock refers to a large enough subset of entities thatmove close to each other for a given time interval. Wepresent a parallel approach, to be run on a Graph-ics Processing Unit, for reporting maximal ocks. Wealso provide experimental results that show the e-ciency and scalability of our approach.

1 Introduction

A trajectory is a sequence of sampled time-stampedpoint locations describing the path of a moving entityover a period of time. Assuming that the movementof an entity between two consecutive positions is doneat constant speed and without changing direction, itstrajectory is modelled by the polygonal line, that canself-intersect, whose vertices are the trajectory points.In this paper we study the ock pattern [1, 2, 3],

that identies a group of entities moving close to-gether during a given time interval (see Figure 1).

e2

e3e0e1

x

y

tτ−1

D1

D2

t0

Figure 1: Example of two ocks.

The increasing programmability and high compu-tational rates of Graphics Processing Units (GPUs),together with CUDA (Compute Unied Device Archi-tecture) and some programming languages which usethis architecture, make them attractive to solve prob-lems which can be treated in parallel as an alternativeto CPUs. GPUs are used in dierent computationaltasks where a big amount of data or operations haveto be done, whenever they can be processed or donein parallel. Some recent works show that demandingalgorithms in dierent elds can take advantage of theGPU parallel processing [4, 5, 6].

∗Email:(mfort,sellares,ivalladares)@imae.udg.edu.Authors research supported by TIN2010-20590-C02-02.

In this paper, we present an ecient parallel GPU-based algorithm, designed under CUDA architecture,for reporting maximal ocks. The experimental re-sults obtained with the implementation of our algo-rithm show the signicance of the presented approachaccording to its performance and scalability.

2 The ock pattern

Let E = e0, . . . , en−1 be a set of n moving entities.The trajectory Ti of the entity ei is a sequence of τpoints in the plane Ti : e

i0, . . . , e

iτ−1, where e

ij denotes

the position of ei at time tj with 0 ≤ j < τ . Weassume that the positions are sampled synchronouslyfor all the entities, and that entity ei moves betweentwo consecutive positions eij , e

ij+1 with constant speed

and without changing direction. Consequently, thetrajectory Ti is described by the polygonal line, whichmay self-intersect, whose vertices are the trajectorypoints.Flocks identify groups of entities whose trajectories

are close together during a minimum period of time.Formally, according to [2], given a set E of entities,a minimum number of entities µ ∈ N, a number oftime-steps δ ∈ N, a time interval Iδj = [tj , tj+δ−1],and a distance ϵ ∈ R:Denition 1 A ock f(δ, µ, ϵ) in a time interval Iδj ,consists of at least µ entities such that for every dis-crete time step tℓ ∈ Iδj , there is a disk of radius ϵ thatcontains all the µ entities.

The number of reported or determined ocks is a crit-ical issue that can adversely aect the response timeand the proper interpretation of the nal results. Fora time-step tℓ ∈ Iδj there may be several circles withradius ϵ that yield a ock with the same subset of enti-ties of E, so we consider that two ocks are dierent ifthey involve two dierent subsets. Since there can beΘ(n2) combinatorially distinct ways to place a circleof radius ϵ among n points in the plane, at each time-step there can be Θ(n2) ock candidates involving dif-ferent subsets of entities [9]. An algorithm having asoutput the position of the entities involved in a ock,would have an output size of Θ(n3) per time-step.Moreover, it is easy to observe that a set of entitiescould be present in many ocks, and even one singleentity can be involved in several ocks. For example, aock of µ+1 entities contains µ+1 ocks of µ entities.

19

Reporting ock patterns

To overcome this problem, as in [3], instead of ndingall the existent ocks, we are interested only in max-imal ocks, so that the entities of one ock may notbe a subset of the entities of another ock. Even inthe case of nding maximal ocks, at every time-stepthere may still asymptotically be Θ(n2) ock candi-dates, although in many real situations the number ofcandidates decreases considerably.

Denition 2 A maximal ockmf(δ, µ, ϵ) in the timeinterval Iδj , is a maximal ock f(δ, µ, ϵ) in Iδj withrespect to the subset inclusion.

We will denote Mδj the family of maximal ocks in

the time interval Iδj , andMδ the family of all maximal

ocks, that is, the ocks of Mδj , for j = 0, . . . , τ − 1.

2.1 Characterization

Vieira et al. [3] prove that, for each pair of entity lo-cations, there exists two such disks. Thus, 2n2 disksper time step must be tested. Since we are interestedin those ocks that are maximal, we prove that keep-ing just one of the two disks is enough. Consequently,only n2 disks per time step must be tested, thus thenumber of tests is reduced by half.The following lemma, similar to one presented in

[3], allows us to bound the number of disks to betested in each time step.

Lemma 3 Let P be a set of n points and D′ be a diskof radius r that covers a subset P ′ ⊆ P , |P ′| = k ≤ n.There exist another disk D′′ of radius r such that: a)has at least two points pi, pj ∈ P ′ on its boundary; b)covers a superset P ′′ of P ′, P ′ ⊆ P ′′ ⊆ P ; c) it islocated at the right side of points pi, pj.

3 Reporting ock patterns

In this paper study the problem of reporting the fam-ily Mδ of all maximal ocks. To report Mδ, wemust nd the family Mδ

j in the time interval Iδj , forj = 0, . . . , τ − 1.

3.1 Computing the family Mδj

Next, we describe the two main steps of the algorithmused to compute the family Mδ

j in the time interval

Iδj .First step. At each time step ti in the time inter-

val Iδj , we compute the family Fi of potential ockscontaining at least µ entities. Each family Fi is ob-tained by using the characterization given in Lemma3. Next, we nd the maximal sets of each family Fi.Abusing of notation, we still denote Fi the obtainedsubfamily of potential maximal ocks. At the endof the step, we have the array of families of poten-tial maximal ocks Fδ

j = [Fj , . . . ,Fj+δ−1] in the time

interval Iδj .

Second step. We rst compute the family Iδj ob-

tained intersecting the δ families of potential maxi-mal ocks in Fδ

j . That is, Iδj contains the sets which

are the intersection of δ sets, one of each family Fi

in Fδj . Finally, we nd the subfamily of maximal sets

of Iδj that contain at least µ elements. The obtained

subfamily is the desired Mδj .

4 GPU implementation

In this Section, we describe how to implement thedierent steps of the algorithm that computes Mδ

j ,when GPU parallel techniques are used.

4.1 The multi-grid structure GWe denote by Ei = e0i , . . . , e

n−1i the set of locations

of the entities of E at time step ti, i ∈ 0, . . . , τ −1. In [3], a single grid is used to perform quick diskrange searching queries over each set Ei. Instead ofusing a grid for each time step, we build a multi-gridstructure, denoted G, in parallel in the GPU, thatallows us to search simultaneously in each one of theδ sets of Eδ

j = [Ej , . . . , Ej+δ−1].The multi-grid structure contains δ regular grids,

where the edges of the grid cells have length ϵ. Eachgrid j contains the set of points corresponding to Ej .In order to maximize the parallel performance, thetop-right corner is dened so that the grid is a square,that is, it has the same number of rows and columns(see Figure 2).The multi-grid structure G is composed of 7 arrays

allocated in the GPU global memory. In Figure 2 wecan see an example for δ = 3. The bottom-left andthe top-right corners of each grid are stored in thearray bδ of size δ, where bδ[j] stores the two cornerpoints of the grid containing Ej . The number of cellsper row (or per column) of each grid is stored in cδand its prex sum is stored in pδ. Both arrays areinteger arrays of size δ.To represent the δ grids, we use the array of inte-

gers cδ, whose size depends on the size of the dierentgrids which directly depend on the distribution of thepoints. For a given grid cell, we store in the corre-sponding cδ position the number of points containedon that cell. The array pδ of the same size of cδ is ob-tained as the prex sum of cδ. Note that, for a betterunderstanding, in Figure 2 the arrays cδ and pδ areshowed as several 2D grids, but in the GPU memorythey are stored one 1D array each, by storing one gridafter the other starting from left to right and from topto bottom, thus, the rst grid corresponds to the timestep ti and the last one of the time step ti+δ. Withthis structure we can easily access, in a parallel way,to any element of the δ grids, by using the fact thatthe set of points of a cell w starts at position pδ[w]of eδ, and is stored in cδ[w] consecutive positions ofeδ. In each eδ position we store the two oats of the

20


corresponding location. Finally, eδ is an integer arrayof size nδ containing at position j the entity id of theeδ[j] location.

cδ

pδ

t0 t1 t2

2

1

1 1

1

3

0 0

0 0

0

0

0

0

0 2

2 2

3 3 4

5 5 5

5 6 6

6

p00

eδ p10 p2

0

t0

cδ

pδ

2 3 1

0 2 5

t0 t1 t2

0eδ 21

p01

p11

p21

p12

p22

p02

021 021

t1 t2

bδ b0 b1 b2

ǫ

e0e1

e2

ǫ

Figure 2: Example of multi-grid G for δ = 3.

We construct the G structure in parallel as follows.We allocate CPU arrays cδ

′, pδ′, b′δ of size δ, and p′

and eδ′ of size nδ. While we load the input P into p′

the arrays cδ′, pδ

′ and b′δ are accordingly lled. Thencδ, pδ, bδ and p are allocated in GPU memory andthe cδ

′, pδ′, b′δ and p′ values are transferred to GPU

memory to cδ, pδ, bδ and p respectively.Then, cδ and pδ are allocated in the GPU memory

according to cδ and pδ values. Additionally, the ar-rays eδ and eδ are allocated with size nδ also in GPUmemory. Because GPU has no dynamic memory wehave to construct G in two steps.First, we ll cδ by counting the number of points

contained in each cell. This is done in parallel bylaunching a kernel with nδ threads, that is, a threadper point and per time step. Each thread idx reads itsp[idx] value and determines its position in the grid byusing its position on the plane, the grid correspondingto the time step which the point belongs to, and the cδand pδ arrays. The corresponding cδ position is incre-mented in 1. Because many threads may correspondto the same cδ position we use atomic operations toensure no thread interferences.Once cδ is computed, pδ is obtained as the prex

sum of cδ and we allocate an auxiliary array v with thesame size of cδ, initialized to zeros. Now, we launcha parallel kernel with nδ threads where each threadidx reads its p[idx] value, determines its position w inthe grid and stores p[idx] at eδ[pδ[w] + v[w]]. Everytime a thread stores its associated point into eδ thecorresponding v value is incremented in one by usingan atomic operation.

4.2 Finding potential ocks

The computation of the array of potential ocks Fδj of

a given interval Iδj is done in two main steps. First, we

need to nd the center of the disks which are obtainedapplying the characterization given in Lemma 3, andsecond, we need to report the entities contained inthose disks. Additionally, because GPUs do not havedynamic memory, we split the process in several steps.

First, we allocate an array of integers Dc of size nδto count how many disks there exist. This is done inparallel by launching a kernel with nδ threads, thatis, a thread per point within Iδj . Using the struc-ture G each thread idx goes through its neighborsand counts the number of points which are at dis-tance ≤ 2ϵ. The results are stored in Dc[idx]. Be-cause this is computed in parallel, each disk will bereported twice, once for each of the two points den-ing the disk. In order to avoid this, we force threadsto check those points whose entity id is bigger thanitself. Then, Dp of size nδ, is computed as the pre-x sum of Dc, and the array D is allocated with sizeDp[nδ−1]+Dc[nδ−1], where we will store the centersof disks. The same process is repeated, but, instead ofcounting, each thread reports the center of the disk.Let us recall that only the right disk per paired pointhas to be reported.

In the second step, we actually report the poten-tial ocks. To do this we use G to perform the rangesearching queries, using the values of the array D asthe center of the queries with radius ϵ. Note that,because in order that the regular grid structure worksfor range searching queries, we must be able to locateany point within the grid. It may happen that somedisk centers were placed outside the grid for a giventime step. Thus, before we perform the range search-ing queries we have to reconstruct G so that the centerof the disks were contained in G. We could modify G,but it is faster to rebuild it than to modify it.

To count the number of points contained on eachdisk we rst allocate Pc, the array of integers of sizeequal to the total number of disks reported which isDp[nδ−1]+Dc[nδ−1], initialize it to zeros and launcha kernel with one thread per disk. Each thread idxreads the center of its disk D[idx], locates D[idx] onthe grid and checks the distance between the centerof the disk and the points of its cell neighbors. Eachthread counts how many points are at distance ≤ ϵ toits disk center, and if there are at least µ the numberis stored in Pc[idx]. Finally, we compute Pp as theprex sum of Pc.

If there is at least one disk per time step with atleast µ entities, we continue to report the potentialocks. The last position of Pp plus the last positionof Pc gives us ω, the size of the array used to store thepotential ocks. Thus, the array P is allocated as aninteger array with size ω and is lled in parallel usinga thread per disk. The process is the same as beforebut this time, instead of counting, each thread storesin P the entities contained inside each disk.

Note that, Pc, Pp and P denote a structure contain-

21

Reporting ock patterns

ing δ families of sets. This is the array of potentialocks Fδ

j , that is, δ families of sets containing the po-tential ocks of δ consecutive time steps starting attime step tj . We denote Fj the family of potentialocks of the time step tj .The last step is to reduce each Fi ∈ Fδ

j so that itonly contains the sets which are maximal, with respectto the partial order induced by the subset relation ineach family. To this end, we use the parallel algorithmof Fort et al. [7]. When the process nishes, we havein Fδ

j the maximal potential ocks with µ or more

entities at each time step of the interval Iδj .

4.3 Reporting Mδ ocks

We want to report all the ocks Mδj for all Iδj , j =

[0, . . . , τ − δ]. Thus, we could perform the same algo-rithm over each Iδj , but many information computedfor a given time step within a time interval can bereused for the following time intervals. The idea isto compute the potential ocks for a given intervaland compute their intersections in a way that we canreuse the information for the next time interval. Tothis aim we proceed as follows.We start at the interval Iδ0 and we compute Fδ

0 .Then, in order to obtain Mδ

0 = Fδ−1∩Fδ−2∩· · ·∩F0

we start at Fδ−1 and we intersect Fδ−2 ∩Fδ−1. Next,we intersect Fδ−3∩Fδ−2∩Fδ−1 and so on. After eachintersection, we compute the maximal sets for eachresulting intersection discarding those sets which arenon maximal or containing less than µ entities. Whenwe are done, we add, to Mδ, the family Mδ

0, whichcontains the ocks of Iδ0 .For the following time steps, instead of recomput-

ing all the potential ocks and all the intersections, wereuse the previous computations. If for a given timestep tj , a family of potential ocks Fk, j ≤ k < j+δ isnot inside the array Fδ

j , we compute Fδj+δ−1. That is,

instead of computing the potential ocks of one singletime step we compute them in δ intervals. Then, weperform the corresponding intersections. This tech-nique does not decrease the number of intersection wehave to do, but, in practice, the δ families we intersectin each step are smaller than the ones containing thepotential ocks, leading to faster results. We repeatthe process until j = τ − δ.The intersection process is also performed in paral-

lel using the GPU algorithm presented by Fort et alin. [8].

5 Experimental results

The experimental results are obtained using an In-tel Core2 CPU 6400 with a Nvidia GTX 480. Wesplit the running time into 2 main steps, the potentialocks reporting process and the intersection process.The accumulated value, corresponding to the columnsheight, gives the total running time of the algorithm.

The algorithm has been tested with a data set ex-tracted from [10], that consists of 145 trajectories of2 school buses around Athens metropolitan area inGreece with a total of 66, 096 time steps.The results (Figure 3) shows that when we vary ϵ

while we maintain µ = 5 and δ = 10, the number ofocks reported increases as long as we increase ϵ. Themost aected part is the intersection process. This isbecause the more ocks we report the more intersec-tions we have to compute. When the parameter µ isvaried, while ϵ = 1200 and δ = 10, the condition toform ock is more restrictive. Thus, the number ofocks decreases and, consequently, the running timestoo.

Figure 3: Running times varying ϵ and µ.

References

[1] J. Gudmundsson, M. J. van Kreveld, B. Speckmann, E-cient Detection of Patterns in 2D Trajectories of MovingPoints, GeoInformatica 11 (2) (2007) 195215.

[2] M. Benkert, J. Gudmundsson, F. Hübner, T. Wolle, Report-ing ock patterns, Computational Geometry 41 (3) (2008)111125.

[3] M. R. Vieira, P. Bakalov, V. J. Tsotras, On-line discoveryof ock patterns in spatio-temporal data, in: D. Agrawal,W. G. Aref, C.-T. Lu, M. F. Mokbel, P. Scheuermann,C. Shahabi, O. Wolfson (Eds.), GIS, ACM, 2009, pp. 286295.

[4] J. D. Owens, D. Luebke, N. Govindaraju, M. Harris,J. Krüger, A. E. Lefohn, T. J. Purcell, A survey ofgeneral-purpose computation on graphics hardware, Com-puter Graphics Forum 26 (1) (2007) 80113.

[5] N. Coll, M. Fort, N. Madern, J. A. Sellarès, Multi-visibilitymaps of triangulated terrains, International Journal of Ge-ographical Information Science 21 (10) (2007) 11151134.

[6] M. Fort, J. A. Sellarès, N. Valladres, Computing popularplaces using graphics processors, in: Proc. SSTDM'10 incooperation with IEEE ICDM'10, IEEE Computer Society,2010, pp. 233241.

[7] M. Fort, J.A. Sellaès, N. Valladares, Finding extremal setson the GPU (Submitted).

[8] M. Fort, J.A. Sellaès, N. Valladares, Intersecting two fami-lies of sets on the GPU (Submitted).

[9] J. Gudmundsson, M. van Kreveld, B. Speckmann, EcientDetection of Motion Patterns in Spatio-Temporal DataSets, in: D. Pfoser, I. F. Cruz, M. Ronthaler (Eds.), GIS,ACM, 2004, pp. 250257.

[10] Y. Theodoridis, R-Tree portal,http://www.rtreeportal.org (2011).

22



Narcís Coll and Marité Guerrieri∗

Geometry and Graphics Group. Universitat de Girona

Abstract

In this paper we propose a new GPU method able tocompute the 2D constrained Delaunay triangulationof a planar straight line graph consisting of points andsegments. The method is based on an incrementalinsertion, taking special care to avoid conicts duringconcurrent insertion of points into the triangulationand concurrent edge ips.

Introduction

The constrained Delaunay triangulation (CDT) is oneof the fundamental topics in Computational Geom-etry and it is used in many areas such as terrainmodelling, nite element method, pattern recogni-tion, path planning, etc. A CDT of a planar straightline graph (PSLG) can be constructed in the follow-ing way: begin with an arbitrary triangulation of thePSLG points; examine each PSLG segment in turn tosee if it is an edge; force each missing PSLG segmentto be an edge; then ip non-locally Delaunay edgesuntil all edges are locally Delaunay with the provisionthat PSLG segments cannot be ipped. There aretwo ways to force a PSLG segment to be an edge ofthe triangulation. The rst consists of deleting all theedges it crosses, then inserting the segment and retri-angulating the two resulting polygons (one from eachside of the segment). The second approach consists ofiteratively ipping the edges that crosses the segmentuntil the segment is an edge. Qi et al. [2] presented aGPU method for computing the CDT of a PSLG. Thismethod has three phases. The rst phase computesthe Delaunay triangulation of the PSLG points, whilethe second inserts the PSLG segments into the tri-angulation using edge ipping and the third ips theremaining non-locally Delaunay edges. Experimentalresults show that the method achieves a signicantspeedup with respect to the best CPU methods. How-ever, since the Delaunay triangulation is constructedusing a digital Voronoi diagram computed on a uni-form grid, the method needs a huge memory space toallocate the grid and is less ecient when the distri-bution of the input points is far from being uniform.

∗Email: (coll,mariteg)@imae.udg.edu. Authors researchsupported by TIN2010-20590-C02-02.

Our approach of computing the CDT on the GPU ex-tends the method presented in [1] by simultaneouslyinserting points and segments into the triangulation.

1 Our approach

Our algorithm is based on an iterative process thatnishes when all PSLG elements (points and seg-ments) are inserted into the Delaunay triangulationand no edge needs to be ipped. In each iteration asmany elements as possible are inserted with the con-dition that only one point can be inserted into onesingle triangle. Each iteration is divided into foursteps: location, where the triangle containing everynon inserted point is determined; insertion, where atmost one point per each triangle is inserted; marking,where some edges of the triangulation are marked tobe a segment or marked to be ipped because theyare crossed by a segment or they are non-locally De-launay; and ipping, where some of the marked edgesare ipped thus avoiding conicts between them.Let n be the number of points andm be the number

of segments. In order to use the GPU's resources aseciently as possible the following arrays allocated inthe global memory are used:Points. Positions (x, y) in 2D. Its rst three posi-

tions corresponds to the three vertices of a large aux-iliary triangle that contains all points. Size n+ 3.Segments. Indices to Points. Each two consecutive

indices correspond to a segment. Size 2m.Inserted-P. Binary ags to determine whether a

point has been inserted or not. Size n+ 3.Inserted-S. Binary ags to establish whether a

segment has been inserted or not. Size m.Triangles. Indices to Points. Each three consec-

utive indices correspond to a triangle. Position zero ofthis array corresponds to the auxiliary triangle. Size3(2n+ 1).Neighbours. Indices to Triangles. Each three

consecutive indices correspond to the current neigh-bours of the triangle. Size 3(2n+ 1).FutureNeighbours. Indices to Triangles. Each

three consecutive indices correspond to the neigh-bours that the triangle will have after executing apoint insertion or a ip which aects its current neigh-bours. Size 3(2n+ 1).

23


ContainingTriangle. Indices to Triangles torecord which triangle contains a point, in the casethat the point has not been inserted, or otherwise, atriangle incident to the point. Initially, all points arecontained in the auxiliary triangle. Size n+ 3.PointToInsert. Indices to Points to record which

the next point to be inserted in each triangle is. Size2n+ 1.Flip. Flags (0, 1, 2, 3) to discern whether an edge

has to be ipped or not. Each three consecutive agscorresponds to a triangle. Flag 0 indicates the edgeis not a segment and does not have to be ipped.Flag 1 indicates the edge is not a segment and has tobe ipped because it is non-locally Delaunay. Flag 2indicates the edge is a segment and does not have to beipped. Flag 3 indicates the edge is not a segment andhas to be ipped because it is crossed by a segment.Size 3(2n+ 1).EdgeToFlip. Flags (0,1,2,3,4,5,6) to record, for

each triangle, which is the edge that will be ipped(0,1,2), or no edge will be ipped (3) or the edge thatwill be ipped by an adjacent triangle (4,5,6). In thelatter case, it is necessary to subtract 4 to determinethe edge that really will be ipped. Size 2n+ 1.Next we explain in detail each step of the algorithm.

1.1 Location step

For each point p the triangle ContainingTriangle[p]is updated as follows: If point P=Point[p] has not yetbeen inserted into the triangulation (Inserted[p] =0), a walking process is launched until the trianglet that really contains P is reached. If P lies on anedge, t is taken as the triangle adjacent to the edgewith the lowest index. Then, ContainingTriangle[p]is updated with t and VertexToInsert[t] is updatedwith p. Note that VertexToInsert[t] can be updatedsimultaneously by distinct processors. In this manner,VertexToInsert[t] contains the last point reaching t.

1.2 Insertion step

For each triangle t, the point of indexp =VertexToInsert[t] is inserted into the tri-angulation as follows: Let pi =Triangles[3t + i](i = 0..2) be the vertex indices of the triangle t.Triangle t will be triangulated to triangle t withvertex indices p0, p1 and p, triangle 2p + 1 withvertex indices p1, p2 and p, and triangle 2p + 2 withvertex indices p2, p0 and p. To properly determinethe neighbours of these three new triangles, this stepneeds to be subdivided into two parts.In the rst part, for each triangle t with a point

to be inserted p, the future neighbours of the trian-gles Neighbours[3t + 1] and Neighbours[3t + 2] areupdated according to the insertion of p in t.In the second part, for each triangle t, if t has

a point p to be inserted, the arrays Triangles,Neighbours and Flip corresponding to the triangles

t, 2p+ 1 and 2p+ 2 are updated according to the in-sertion of p in t. Otherwise, the neighbours of t aresimply updated with the triangles previously storedin FutureNeighbours[3t..3t+ 2].

foreach p < n+ 3 do /* in parallel */

if Inserted [i] = 0 then

t =ContainingTriangle[p];P =Point[p];Found=false;while Found=false do

[P0, P1, P2] = Points[Triangles[3t..3t+ 2]];if P contained in some segment PjPj+1

and Neighbours[3t+ j]<t thenFound=true; t=Neighbours[3t+ j];else if P contained in triangle P0P1P2

then Found=true;else

C = (P0 + P1 + P2)/3;Seek for j such that segment Cpintersects segment PjPj+1;t =Neighbours[3t+ j];

ContainingTriangle[p]=t;VertexToInsert[t]= p;

Algorithm 1: Location

foreach t < 2n+ 1 do /* in parallel */

p=VertexToInsert[t];if i = −1 then foreach j = 1..2 do

t′=Neighbours[3t+ j];j′=opposite(j);FutureNeighbours[3t′ + j′]=2p+ j;

Algorithm 2: Insertion. Part 1.


p=VertexToInsert[t];if i = −1 then

Inserted-P[i]=1;[p0, p1, p2]=Triangles[3t..3t+ 2];[n0, n1, n2]=FutureNeighbours[3t..3t+ 2];[f0, f1, f2]=Flip[3t..3t+ 2];Triangles[3t..3t+ 2]=[p0, p1, p];Triangles[3(2p+ 1)..3(2p+ 1) + 2]=[p1, p2, p];Triangles[3(2p+ 2)..3(2p+ 2) + 2]=[p2, p0, p];Neighbours[3t..3t+ 2]=[n0, 2p+ 1, 2p+ 2];Neighbours[3(2p+ 1)..3(2p+ 1) + 2]=[n1, 2p+ 2, t];Neighbours[3(2p+ 2)..3(2p+ 2) + 2]=[n2, t, 2p+ 1];Flip[3t..3t+ 2]=[f0,−1,−1];Flip[3(2p+ 1)..3(2p+ 1) + 2]=[f1,−1,−1];Flip[3(2p+ 2)..3(2p+ 2) + 2]=[f2,−1,−1];

else

Neighbours[3t..3t+ 2]=FutureNeighbours[3t..3t+ 2];

Algorithm 3: Insertion. Part 2.

24


1.3 Marking step

In this step the edges corresponding to a PSLGsegment and the candidate edges to be ipped aremarked. An edge can be a candidate due to beingcrossed by a segment or being non-locally Delaunay.Accordingly, this step is subdivided into two parts:In the rst part, for each non inserted segment s

whose both endpoints have already been inserted intothe triangulation, if the endpoints of s are connectedby an edge, this edge is marked as constrained. Oth-erwise, a walking process starting from the rst end-point is launched until a ippable edge crossed by sand not ipped in the previous ipping step is found.Then, this edge and its opposite edge are marked tobe ipped. The same is done starting from the otherendpoint.

foreach s < m do /* in parallel */if Inserted-P[s]=0 andInserted-P[Segments[2s]]=1 andInserted-P[Segments[2s+ 1]]=1 then

for j = 0..1 do

if j = 0 then

[i0, i1]=Segments[2s..2s+ 1];else [i1, i0]=Segments[2s..2s+ 1];[P0, P1]=Points[i0..i1];starting from ContainingTriangle[i0] seekfor the triangle of index t incident to i0intersecting segment P0P1;if P1 is a vertex of t then

j=edge of t connecting P0 and P1;t′=Neigbours[3t+ j];j′=opposite(j);Flip[3t+ j]=Flip[3t′ + j′]=2;Inserted-P[s]=0;

else

Found=false;while Found=false do

j=edge of t intersecting segmentP0P1;if j is ippable and dierent toEdgeToFlip[t] then

Found=true;t′=Neigbours[3t+ j];j′=opposite(j);Flip[3t+ j]=Flip[3t′ + j′]=3;

else

t=Neigbours[3t+ j];

Algorithm 4: Marking. Part 1.

In the second part, for each triangle t not crossedby any segment (Flip[3t] mod 2+Flip[3t + 1] mod2+Flip[3t + 2] mod 2=0) the non constrained edgesthat are non Delaunay are marked to be ipped.

1.4 Flipping step

During this step, at most one edge of each triangle tis ipped. To avoid conicts between concurrent ips,


if∑2

k=0(Flip[3t+ k] mod 2) = 0 then for

j = 0..2 do

if Flip[3t+ j]= 2 and no Delaunay(j) thenFlip[3t+ j]=1;

Algorithm 5: Marking. Part 2.

this step needs to be subdivided into three parts.In the rst stage, a marked edge whose opposite

edge is the only marked edge in its triangle t′ is sought.If this edge exists, let nc be the number of markededges of t. Then, if nc ≥ 2 or (nc = 1 and t < t′) theedge is stored in EdgeToFlip[t].In the second stage, if t has an edge to be ipped

(EdgeToFlip[t]≤ 2), the future neighbours of the tri-angles adjacent to the quadrilateral determined by theedge are updated accordingly to the future ip of theedge. Otherwise, if the opposite edge of an edge j oft is to be ipped, 4 + j is stored in EdgeToFlip[t].In the third stage, if t has an edge to be

ipped (EdgeToFlip[t]≤ 2), the arrays Triangles,Neighbours and Flip corresponding to the trianglest and t′ are updated according to the ip. Otherwise,if any adjacent triangle has an edge to be ipped, thearray Neighbours is updated.


EdgeToFlip[t]=3;

nc=∑2

k=0(Flip[3t+ k] mod 2);if nc > 0 then

j = 0;Found=false;while j ≤ 2 and no Found do

if Flip[3t+ j]=1 then

t′=Neigbours[3t+ j];j′=opposite(j);

nc′=∑2

k=0(Flip[3t′ + k] mod 2);

if Flip[3t′ + j′]=1 and nc′ = 1 and(nc ≥ 2 or (nc = 1 and t < t′)) then

EdgeToFlip[t]=j;Found=true;

j ++;

Algorithm 6: Flipping. Part 1.

2 Results

The algorithm was implemented using OpenCl ona computer equipped with an Intel(R) Pentium(R)D CPU 3.00GHz, 3,5GB RAM and a GPU NVidiaGeForce GTX 580/PCI/SSE2. Each one of the al-gorithm's parts was written in a kernel. The algo-rithm was executed ten times on two dierent models(Chairs32×32 and Holland4×4) and compared withTriangle [3], one of the most popular computationalgeometry software. The model Chairs32×32 is formedby 1024 copies of the model showed in the Figure

25



if EdgeToFlip[t]≤ 2 then

j=EdgeToFlip[t]; j1=j + 1 mod 3;n=Neighbours[3t+ j1]; nj1=opposite(j1);FutureNeighbours[3n+ nj1]=t′;t′=Neighbours[3t+ j]; j′=opposite(j);j1′=j′ + 1 mod 3;n′=Neighbours[3t′ + j1′]; nj1′=opposite(j1′);FutureNeighbours[3n′ + nj1′]=t;

else

for j=0..2 do

t′=Neighbours[3t+ j];if EdgeToFlip[t′]≤ 2 andNeighbours[EdgeToFlip[t′]]=t thenEdgeToFlip[t]=4 + j;



if EdgeToFlip[t]≤ 2 then

j=EdgeToFlip[t];t′=Neighbours[3t+ j]; j′=opposite(j);j1=j + 1 mod 2; j2=j + 2 mod 2;j1′=j′ + 1 mod 2; j2′=j′ + 2 mod 2;p2=Triangles[3t+ j2]; p3=Triangles[3t′ + j2′];Triangles[3t+ j1]=p3; Triangles[3t′ + j1′]=p2;ContainingTriangle[p3]=t;ContainingTriangle[p2]=t′;n1=FutureNeigbours[3t+ j1];n2=FutureNeigbours[3t+ j2];n1′=FutureNeigbours[3t′ + j1′];n2′=FutureNeigbours[3t′ + j2′];Neigbours[3t+ j]=n1′; Neigbours[3t+ j1]=t′;Neigbours[3t+ j2]=n2;Neigbours[3t′ + j′]=n1;Neigbours[3t′ + j1′]=t;Neigbours[3t′ + j2′]=n2′;FutureNeighbours[3t..3t+ 2]=Neighbours[3t..3t+ 2];FutureNeighbours[3t′..3t′ + 2]=Neighbours[3t′..3t′ + 2];if Flip[3t+ j]=3 then EdgeToFlip[t]=j1;else EdgeToFlip[t]=3;

else if EdgeToFlip[t]= 3 then

Neighbours[3t..3t+ 2]=FutureNeighbours[3t..3t+ 2];

else

j=EdgeToFlip[t]-4; j1=j + 1 mod 2;if Flip[3t+ j]=3 then EdgeToFlip[t]=j1;else EdgeToFlip[t]=3;

for j=0..2 do if Flip[3t+ j]= 2 then

Flip[3t+ j]=0;


1 arranged in a 32 by 32 array, while the modelHolland4×4 is formed by 16 copies of the modelshowed in the Figure 2 arranged in a 4 by 4 array.The mean running times are presented in Table 1.Our future task is to study the performance of ouralgorithm versus [2]. However, our approach does not

make use of the huge memory space needed by [2] tostore the required digital Voronoi diagram.

Chairs32×32 Holland4×4Num. Vertices 1444864 1015344Num. Segments 739328 1007488Mean time (ms) 4028 2306Triangle (ms) 12835 10254

Table 1: Behaviour of the proposed algorithm.

Figure 1: Cell of the model Chairs32×32

Figure 2: Cell of the model Holland4×4

References

[1] N. Coll, M. Guerrieri, Parallel Delaunay triangulationbased on Lawson's incremental insertion, in: Proceed-ings of the XIV Spanish Meeting on ComputationalGeometry, CRM Documents, 8, Centre de RecercaMatemàtica, Bellaterra (Barcelona), 2011, 169172.

[2] M. Qi, T. Cao, T. Tan, Computing 2D constrainedDelaunay triangulation using the GPU, in: Proceed-ings of the ACM SIGGRAPH Symposium on Inter-active 3D Graphics and Games, I3D '12, ACM, NewYork, NY, USA, 2012, 3946.

[3] http://www.cs.cmu.edu/ quake/triangle.html.

26


Metaheuristic approaches for the Minimum Dilation Triangulation

problem

Maria Gisela Dorzán∗1, Mario Guillermo Leguizamón†1, Efrén Mezura-Montes‡2, and Gregorio Hernández§3

1Universidad Nacional de San Luis - Argentina2Universidad Veracruzana - México

3Universidad Politécnica de Madrid - España

Abstract

We focus on the development of approximated algo-rithms to nd high quality triangulations of minimumdilation because the complexity status of the Min-imum Dilation Triangulation problem for a generalpoint set is unknown. We propose an operator togenerate the neigborhood which is used in dierentalgorithms: Local Search, Iterated Local Search, andSimulated Annealing. Besides, an algorithm calledRandom Local Search is presented where good andbad solutions are accepted using the previous men-tioned operator. We use the Sequential ParameterOptimization method for tuning the parameters of theSA algorithm. We compare our results with the onlyavailable algorithm found in the literature that usesthe obstacle value to sort the edges in the construc-tive process. Through the experimental evaluationand statistical analysis, we assess the performance ofthe proposed algorithms using this operator.

Introduction

Dierent measures are adopted to design optimal tri-angulations. The most popular are the weight, stab-bing number, area, and dilation. Although the us-age of approximated approaches to solve complex op-timization problems is common nowadays, this lastmeasure has not been used as selection criterion whenoptimizing triangulations by using metaheuristic algo-rithms [10].Let S be a nite planar point set, T a triangulation

of S and u, v two points in S. There are two distancemetrics: the Euclidean distance between u and v, |uv|,

∗Email: [email protected]. Research supported byCONICET and Research Project No. 22/F014

†Email: [email protected]. Research supported by Labora-torio de Investigación y Desarrollo en Inteligencia Computa-cional (LIDIC)

‡Email: [email protected]. Research supported by CONA-CyT bilateral project No. 164626.

§Email: [email protected]. Research supported by ESFEUROCORES programme EuroGIGA - ComPoSe IP04 -MICINN Project EUI-EURC-2011-4306

and the length of the shortest path between u and vwith in the triangulation T , distT (u, v). The dilationbetween u and v with respect to T is the ratio betweenthe shortest path and the Euclidean distances between

u and v, and is dened as ∆T (u, v) =dist(u,v)

|uv| .

The maximum over all the dilations between pairsof points in T is called the dilation of T (or stretchfactor) and is represented by ∆(T ). The best possibledilation of any triangulation of S is the dilation of Sand is denoted by ∆(S). Thus, we have

∆(T ) = maxu,v∈S

∆T (u, v) and ∆(S) = minTofS

∆(T )

The triangulation T ∗ whose dilation is the dilationof S, i.e. ∆(T ∗) = ∆(S), is called minimum dila-tion triangulation of S. Note that there are severaltriangulations of minimum dilation for a given set ofpoints.

Computing the minimum dilation triangulation fora set of points in the plane is listed as an Open Prob-lem in Eppstein's survey [6], i.e., no polynomial algo-rithm that can build it is known and no proof thatthe problem is NP-hard is shown. Neither the greedynor the delaunay triangulation algorithms generatethe minimum dilation triangulation of a planar pointset, as shown in Figure 1. Therefore, one approach isto use metaheuristic techniques for obtaining approx-imate solutions to the optimum.

(a) GT (b) DT (c) MDT

Figure 1: A point set for which the MDT neither isthe GT nor the DT.

27

Metaheuristic approaches for the MDT problem

1 Approximated algorithms for

the MDT problem

A set of simple techniques is presented because, to thebest of the authors' knowledge, there are no works inthe literature where the MDT problem is approachedusing metaheuristics. We present the general overviewof the studied algorithms: Greedy [5], Local Search[1, 11], Iterated Local Search [9], Simulated Anneal-ing [3, 7], and Random Local Search describing theirfeatures and parameters. For all the algorithms thesolution space E is represented by all possible trian-gulations of a set S of n points in the plane. An n×nmatrix of ones and zeros is used to represent a possi-ble solution. A 1 at position (i, j) means that thereis an edge connecting points i and j, otherwise a 0 isplaced. The objective function, f : E → R, assigns areal value to each element of E . For each E ∈ E , thefunction f is dened as the maximum dilation amongall pairs of points in E.

Greedy Algorithm (G-MDT) It starts with anempty solution Sol and inserts edges until a triangu-lation for a given set of point S is generated. Let Abe the set of all possible edges that have not been in-serted in Sol, u, v ∈ S and e = uv ∈ A. If the dilationof the points u and v determines the dilation of thesolution, i.e., ∆Sol(u, v) = ∆(Sol), then the edge e isinserted in the solution Sol. This reduces the dilationof the points u and v to 1 because dist(u, v) = |uv|and consequently decreases the dilation of the solu-tion Sol. The insertion is performed whenever thenew edge to insert produces no intersections with theedges already inserted.

Local Search Algorithm (LS-MDT) It startsfrom a random initial solution and then iterativelymoves to a neighbor solution. These moves are per-formed by applying an operator looking for a bettersolution. If a better solution is found, it replaces thecurrent solution by the new one and it continues theprocess until it can not improve the current solution.When no improved solutions are present in the neigh-borhood, local search is stuck at a local optimal so-lution. The moves in the search space are performedusing the retriang() operator that works as follows.Let a, b ∈ S, if the dilation of the points a and b de-termines the dilation of the current solution Sol, i.e.,∆Sol(a, b) = ∆(Sol) (see Figure 2(a)), then a and bare joined by an edge. The egdes intersected by abare deleted and the region delimited by these edges(see Figure 2(b)) is retriangulated in a greedy way.The edge whose points have the higher dilation is in-serted at each step if it does not intersect with theedges previously added and if a complete triangula-tion is not achieved (see Figure 2(c)). Therefore the

current solution is improved because its dilation hasbeen reduced. Due to the behavior of this operatoronly a single neighbor is obtained.

g

f

e

cb

d

a

(a)

g

f

e

cb

d

a

(b)

g

f

e

cb

d

a

(c)

Figure 2: retriang() operator. In (a), ∆Sol(a, b) =∆(Sol) is shown in dashed style. In (b), the edge abis added and the dashed edges intersected by ab aredeleted. The grey region is retriangulated in a greedyway (c).

Iterated Local Search Algorithm (ILS-MDT)First, a local search is applied to an initial randomsolution and yields a local optimum using the LS-MDT algorithm. At each iteration, a perturbation ofSol is carried out and then, a local search is appliedto the perturbed solution Sol′. This process iteratesuntil the number of perturbations is less than 50 andthe current solution is improved, always keeping thebest solution found so far Solbest. The perturb(Sol)procedure is performed by the perturbation operatorwhere n/5 random local retriangulations over randomselected points of S are performed worsening the cur-rent solution, i.e., a point b ∈ S is randomly chosenand all the points adjacent to b form the grey regionthat has to be retriangulated. The edges belonging tothe region are deleted and then this region is retrian-gulated in a random way. The edges of the region areinserted at random if they do not intersect with theedges previously added and if a complete triangula-tion is not achieved.

Simulated Annealing Algorithm (SA-MDT)From an initial solution, it executes a number of iter-ations where a neighbor of the current solution Sol isgenerated in each one of them. The moves that im-prove the cost function are always accepted. Other-wise, the neighbor Sol′ is selected with a given prob-ability that depends on the current temperature T .This probability is usually called acceptance functionand it is evaluated according to p(T, Sol, Sol′) = e−

δT

28


where δ = f(Sol′) − f(Sol). M(Tk) moves are per-formed for each temperature Tk. Finally, the value ofTk is decreased at each algorithm iteration k. The al-gorithm continues this way until the termination con-dition is met.

We use two operators to obtain the neighborhood ofa solution: 1) Local random retriangulation (RR) asused in the perturbation operator in ILS-MDT algo-rithm; 2) Local greedy retriangulation (GR) is similarto the RR described above, but the edges are insertedin a greedy way. A point b ∈ S is randomly chosenand all the points adjacent to b are recovered. Thenthe grey region that forms the adjacent points to bis retriangulated using at each step the edges whosepoints have the higher dilation (if it does not inter-sect with those previously added). The scheduling ofapplication for these operators is as follows. GR isperformed g times and RR is performed r times. Weconsidered g ∈ 4, 6, 8 and r = 2 as will be describedlater in the next section.

Due to the complexity involved in tuning the pa-rameters of metaheuristic techniques, we used Sequen-tial Parameter Optimization (SPO) [2] for tuning theparameters required by SA. All SPO-tuning exper-iments for the SA-MDT algorithm were performedwith the following settings: 50 sequence steps, 5 newdesign points in each step, up to 2 repeats per designpoint, and 7 initial design points. Random Forest wasused as a fast surrogate model building tool. Latinhypercube sampling was chosen as the generator ofdesign points. We obtained better results with Tk

moves at each temperature (M(Tk) = Tk), using lo-cal retriangulation interleaving with g = 8 and r = 2and with a initial temperature equal the maximumlength of the paths in the initial solution.

Random Local Search Algorithm (RLS-MDT)Initially, the proposed SA algorithm used the operatorretriang() to move in the solution space. We observedthat such algorithm converged too early to suboptimalregions (the best solution was achieved during the rsttemperature value). Therefore we proposed anotheroperator for the SA algorithm to avoid this conditionof premature convergence. On the other hand, the re-sults obtained with operator retriang() were encour-aging with the LS and ILS algorithms. Therefore, wepropose a new algorithm; we called it Random LocalSearch algorithm, which starts with a random initialsolution. maxEvals = 2n evaluations are performed,accepting good and bad solutions. This mechanismallows to explore and exploit the search space, alwayskeeping the best solution found so far Solbest. Weconsider the operator retriang() but the retriangula-tion of the region is calculated in a random mannerbecause better results were obtained that way.

2 Experimental Evaluation and

Statistical Analysis

We generated 20 problem instances, each one is calledn-i, where n = 40, 80, 120, 160, 200 is the size ofthe instance and i is the number of the instance(1 ≤ i ≤ 4). The points are randomly generated,uniformly distributed and for each point (x, y), thecoordinates x, y ∈ [0, 1000]. For stochastic algorithms25 runs were carried out with dierent initial seedsand dierent initial triangulations for each run.Table 1 shows the best values obtained with the

proposed algorithms: G-MDT, LS-MDT, ILS-MDT,SA-MDT, and RLS-MDT. The lowest dilations arebolded. Xia proved that the dilation of the Delau-nay triangulation of a set of points in the plane isless than 1.998 [12]. All the results obtained by thealgorithms are less than 1.998 and the Delaunay trian-gulation never obtains better results for the instancesconsidered (see the DT column). We compared ourresults with those obtained with the algorithm pro-posed in [8]. We called it OV-MDT because it com-putes the Obstacle Value of the edges for approximat-ing the minimum dilation triangulation. We used theJava-applet available at the Geometry Lab site at theUniversity of Bonn (www.geometrylab.de) to obtainthe results. For some instances the applet did notproduce a result giving an error and halting the exe-cution of the algorithm. It should be noted that theapplet yielded results after several days of execution(see the OV-MDT column). TheG-MDT algorithmobtained solutions of poor quality even for the small-est instances. The RLS-MDT algorithm obtained thelowest (best) dilations for all problem instances (ex-cept for one instance of 120 points). We observedthat the OV-MDT algorithm did not obtain betterresults than RLS-MDT. Although in some instancesequal best values were obtained by some algorithms,the RLS-MDT algorithm showed to be less complexand faster than the other algorithms.As all samples had a non-Normal distribution

(Kolmogorov-Smirnov test) we used non-parametricstatistical tests to evaluate the condence on the sta-tistical results provided by the algorithms. In orderto carry out a comparison which involved results frommore than two algorithms Kruskal Wallis test wasused for multiple comparisons with independent sam-ples [4]. We used the post-hoc Tukey test to ndwhich algorithms' results are signicantly dierentfrom one another. These tests are well-known andthey are usually included in standard statistics pack-ages (such as Matlab, R, etc.). After applying theKruskal Wallis test, we observed that there was al-ways a signicant dierence between the algorithms(1.e-16 < p-value≤ 0.0015) being this dierence muchlarger when considering sets with more points. TheILS-MDT and RLS-MDT algorithms had similar per-

29

Metaheuristic approaches for the MDT problem

Instance DT OV-MDT G-MDT LS-MDT ILS-MDT SA-MDT RLS-MDT

40-1 1.31871 132.863 1.37203 130.959 1.29467 1.29467 1.29467

40-2 1.37491 1.36881 1.37491 1.36881 1.36881 1.36881 1.36881

40-3 1.32268 1.32268 1.36026 1.32268 1.32268 1.32268 1.32268

40-4 1.35391 1.32330 1.34919 1.32330 1.32330 1.32557 1.32330

80-1 1.33034 1.32363 1.39394 1.32363 1.32363 1.40844 1.32363

80-2 1.40500 135.181 1.38199 1.35339 1.32418 1.50331 1.32418

80-3 1.37791 1.30519 1.36180 1.34065 1.31457 1.37791 1.30519

80-4 1.42547 134.239 1.39362 1.33176 1.31583 1.47561 1.31583

120-1 1.37428 1.34442 1.42386 1.35398 1.34825 1.72082 1.34442

120-2 1.33973 131.194 1.37778 1.31194 1.30366 1.65921 1.31194120-3 1.37724 134.016 1.43027 1.40314 1.31985 1.74912 1.29786

120-4 1.38529 1.33600 1.39972 1.35152 1.34272 1.68163 1.33600

160-1 1.34365 1.31050 1.43076 1.35236 1.33384 1.95530 1.31050

160-2 1.35409 NA 1.33260 1.36463 1.33260 1.97896 1.33260

160-3 1.35797 133.278 1.37723 1.37178 1.36352 1.87574 1.32995

160-4 1.37922 1.34941 1.37922 1.37922 1.35037 1.87033 1.34941

200-1 1.35751 NA 1.42220 1.41867 1.35751 2.08064 1.35751

200-2 1.39512 NA 1.39512 1.36451 1.36350 2.06324 1.36350

200-3 141.962 NA 1.45763 1.40645 1.40645 2.09818 1.40645

200-4 1.36965 NA 1.40765 1.35499 1.35251 2.00110 1.35251

Table 1: Best results of DT, OV-MDT, G-MDT, LS-MDT, ILS-MDT, SA-MDT, and RLS-MDT algorithms.

formances for the whole set of instances because therewere no signicant dierences between these two al-gorithms (using the Tukey test).

3 Conclusions

In this work four metaheuristic techniques wereimplemented to nd high quality triangulationsof minimum dilation. The set of instancesgenerated and used in the experimental evalu-ation are available at the research project site(www.dirinfo.unsl.edu.ar/bd2/GeometriaComp/)

After performing the experimental evaluation andstatistical analysis, we assessed the applicability ofthe LS, ILS, SA, and RLS algorithms for the MDTproblem. The RLS-MDT and ILS-MDT algorithmshad similar competitive performances with respect tothe other algorithms in the problem instances consid-ered. These algorithms achieved a dilation reductionbetween 0.4% and 9.2% with respect to the greedystrategy (GT-MDT ). Although both algorithms be-have similarly, the RLS-MDT algorithm required lessevaluations than ILS-MDT. It should be noted thatthe existing algorithm (OV-MDT ) yielded no resultsfor four of the instances considered as either it re-turned an error or halted. The RLS-MDT algorithmoutperformed the OV-MDT algorithm in 50% of theinstances considered and obtained the same results inthe other instances.

References

[1] E. Aarts and J. Lenstra. Local Search in Combinato-rial Optimization. John Wiley, 1997.

[2] T. Bartz-Beielstein. Experimental Research in Evo-lutionary Computation: The New Experimentalism(Natural Computing Series). Springer, 2006.

[3] V. Cerný. Thermodynamical approach to the travel-ing salesman problem: An ecient simulation algo-rithm. Journal of Optimization Theory and Applica-tions, 1985.

[4] J. Derrac, S. García, D. Molina, and F. Herrera. Apractical tutorial on the use of nonparametric statis-tical tests as a methodology for comparing evolution-ary and swarm intelligence algorithms. Swarm andEvolutionary Computation, 1(1):3 18, 2011.

[5] J. Edmonds. Matroids and the greedy algorithm.Mathematical Programming, 1(1):127136, 1971.

[6] D. Eppstein. Spanning trees and spanners. In J.-R.Sack and J. Urrutia, editors, Handbook of Computa-tional Geometry, chapter 9, pages 425461. Elsevier,2000.

[7] S. Kirkpatrick, C. Gelatt, and M. Vecchi. Optimiza-tion by simulated annealing. Science, 220:671680,1983.

[8] A. Klein. Eziente berechnung einer dilationsmini-malen triangulierung. Master's thesis, 2006.

[9] O. Martin, S. Otto, and E. Felten. Large-step markovchains for the traveling salesman problem. ComplexSystems, 5:299326, 1991.

[10] Z. Michalewicz and D. Fogel. How to Solve It: Mod-ern Heuristics. Springer, 2004.

[11] C. Papadimitriou. The complexity of combinatorialoptimization problems. PhD thesis, Princeton, NJ,USA, 1976. AAI7704795.

[12] G. Xia. Improved upper bound on the stretch factorof delaunay triangulations. In Proceedings of the 27thannual ACM symposium on Computational geometry,SoCG '11, pages 264273, New York, NY, USA, 2011.ACM.

30


Three location tapas calling for CG sauce

Frank Plastria∗1

1MOSI - Vrije Universiteit Brussel,Pleinlaan 2, B 1050 Brussels, Belgium

Abstract

Based on some recent modelling considerations in lo-cation theory we call for study of three CG constructsof Voronoi type that seem not to have been studiedmuch before.

Introduction

There is a strong interrelation and mutual ensemina-tion between (continuous) location theory and com-putational geometry. The first generates interestinto distance optimization problems with geometricalinterpretations, the second builds geometrical algo-rithms for efficient solution to the first’s problems.In this talk I will shortly present some recent work

stemming from location theory calling for study ofthree novel (?) computational geometry constructs.

1 Mixed norm shortest paths

The fact that the distance measure may be differentfrom one region to another, contrary to the usual as-sumption that distance is measured by a single norm,has been acknowledged in only a few location stud-ies. Parlar [14] considers the plane divided by a linearboundary with at one side `2 and at the other side `1.Brimberg et al [2] consider a bounded region with adifferent norm inside and outside, focusing in partic-ular on an axis-parallel rectangular city with `1 insideand `2 outside.Brimberg et al. [1] and Zaferanieh et al. [19] con-

sider location in a space with two distinct `p norms incomplementary halfplanes. The way to calculate thedistance in such a space was studied more in detail byFranco et al [7].Fathali and Zaferanieh [5] extend this work to in-

clude more general block norms. Fliege [6] consid-ers differentiably changing metrics similar to Riemannspaces.Unfortunately part of this work is wrong. All au-

thors consider only two possibilities when calculatingdistances: when the two points lie in the same half-plane simply use the corresponding distance, and oth-

∗Email: [email protected]

erwise in two steps via the best possible point on theseparating line. Although this is true in some partic-ular cases, e.g. the axis-parallel rectangular city caseevoked before (but without inflation factors), Parlaralready observed that in general when calculating dis-tance in this naive way distance to a fixed point isnot continuous everywhere. Worse: triangle inequal-ity may be violated. This clearly shows that suchdistance calculation cannot be correct, but none ofthe authors try to resolve this discrepancy.What should rather be done is to consider shortest

path distance in the space, similar to what is donein the so-called weighted (euclidean) region problemwell known in CG since the original paper of Mitchelland Papadimitriou [10] (which depicts a clear coun-terexample to the naive distance above): the length ofthe shortest possible piecewise linear path using theadequate measure (speed in that paper) in each piece.For two halfspaces with arbitrary distinct norms or

gauges we studied more in detail the optimality condi-tions when crossing the boundary, generalizing Snell’slaw in optics. This has nice geometric interpretationand leads to a geometrical view on deriving such dis-tances [16].What turns out to be crucial is the comparison be-

tween the two norms along (the direction of) the sep-arating line. As long as these are equal things remainrelatively simple and the naive assumption about dis-tances is correct. However, when they differ thingschange. One region is then ‘slower’ than the other(as measured along the boundary line). And in sucha case some paths connecting two points within theslowest region will consist of three pieces. They ‘hitcha ride’ along the quicker boundary.Thereby continuity of the distance is again guaran-

teed, but now convexity is partially lost, as illustratedby the balls in the figure below. At right of the verticalseparation line we have the slower region in gray withdistance measure 4`1, at left the faster region withnorm `2. The figure shows balls of increasing mixed-norm radius centered at the dot. For very small radiuswe have the diamond-shaped `1 ball. As soon as theradius allows to reach the boundary a part of circularshape arises at left due to two piece shortest paths,which spills over with a linearly moving front at theright corresponding to three piece paths. The whiteline shows the set of meeting points where one-piece

31

Three tapas with CG sauce

and three-piece paths yield equal distance.

slow: 4`1

fast: `2

It should be noted that Brimberg et al [2] correctlyprove convexity of the distance from a fixed point toall points of the other region, but do not acknowl-edge that this convexity (quite crucial for their sub-sequent optimization approaches) does not extend tothe whole plane if the fixed point lies in the slowerregion.The linearly separated two-norm situation is of

course only the first step. For adequate descriptionof some reality one will have to consider a planesplit into cells each with their own norm or gauge(for asymmetry). How to efficiently calculate shortestpath distances in such a context seems to be largelyopen, apart from some work on approximations, seeCheng et al. [3]. Correctly attacking location prob-lems in such mixed norm spaces is another matterall together. A first step will probably be to look atVoronoi diagrams in such environments. Clearly a lotof opportunities for CG.Interesting applications may be found in fire-

fighting using models to simulate the quite differentways fires spread in various circumstances, taking intoaccount vegetation like (ir)regularly spaced planta-tions of varying types, influence of terrain inclination,and weather conditions, such as winds and humidity,and so forth.

2 Knapsack Voronoi diagramsLet S be a finite set of sources s ∈ R2 with capacitiescs > 0. Consider the location of a central point xin the plane that should be connected to sufficientlymany of the sources to be able to supply given demandD. The supply-weighted sum of distances should be

minimized, possibly together with some additionalcosts f(x):

min∑s∈S

wsd(s, x) + f(x) (1)∑s∈S

ws ≥ D (2)

0 ≤ ws ≤ cs (3)x ∈ R2 (4)

This is a continuous single facility location-allocationproblem that I have been studying under several vari-ants for f(x), most of the time consisting of fixedweighted distances to demand point(s) [9, 17, 18, 15].For any fixed location x finding the best allocation

W = (ws)s∈S for the supplies is a continuous knap-sack problem, easily solved by taking sources s ∈ Sat their full capacity ws = cs in non-increasing or-der of distance d(s, x), until the demand D is met.Only the last chosen source may contribute below itscapacity, and all further sources will not contributeat all (ws = 0). There are only a finite number ofallocations W of such type possible.For any fixed allocation W finding the correspond-

ing best site x amounts to solve a single facilityminisum location problem (Fermat-Weber problem)which may be easily done by various methods of con-vex optimization. And this x should then yield W asoptimal allocation.It is therefore of interest to know the regions with

fixed allocation.The planar subdivision corresponding to different

solutions to this knapsack problem is what I call aKnapsack Voronoi diagram. In case all capacities areequal c, we obtain a traditional k-th order Voronoidiagram with k = dD/ce , with additional splits ofsome cells as soon as demand is not a multiple of ca-pacity. When capacities differ the order k is not fixedand we have new types of diagrams. In particular ver-tices of the diagram may have from 3 up to 6 edges.The following examples show how vertices with 5 or 6edges may arise. Demand is always D = 8, but in thefirst case ca = 10, cb = 6, cc = 5, while in the secondca = 4, cb = 6, cc = 5.

AAAAAAA

a8

abc

cba

c5b3

acb

cab

c5a3

bacb6a2

bca

b6c2

med(a,b)

med(b,c)

med(a,c)

32


AAAAAAA

AAAAAAA

abc

a4b4

cba

c5b3

acb

a4c4

cab

c5a3

bacb6a2

bca

b6c2

med(a,b)

med(b,c)

med(a,c)

As long as the coefficients of the distances d(s, x)in (1) are simply equal to ws the edges of the cor-responding Knapsack Voronoi diagram remain linearsegments. But in some models such as the locationof an assembly station [18] additional factors appearand then we need diagrams similar to multiplicativelyweighted Voronoi diagrams, where cells have circulararcs as boundaries and may be disconnected.This shows the interest of studying the properties

and efficient ways to calculate such diagrams, clearlya task for CG.More general constructs, such as multi-facility ver-

sions of previous location models may be consideredwhere the allocation problem for fixed location(s) ofthe central facility(ies) to be located consists of a lin-ear program with coefficients either fixed or depend-ing only on the distances between sources and facilitysite(s). Such an LP can generically have only a finitenumber of optimal solutions, each corresponding tocertain linear inequalities between the distances, socorresponding to (often empty) cells of a Voronoi-likediagram. I do not know of any study of such struc-tures.

3 Push-pull Voronoi diagramsfor points and polygons

Push-pull location problems try to find good siteswithin a given region for facilities that at the sametime are far (pushed away) from some repelling pointsr ∈ R and close (pulled) to some other attractingpoints a ∈ A.For euclidean distances Ohsawa [11] fully constructs

the set of efficient points for this biobjective maximin-minimax problem. He shows that any such efficientpoint must lie on the boundary of cells obtained byintersecting the farthest-point Voronoi diagram w.r.t.R with the closest-point Voronoi diagram w.r.t. Awithin the given region. These line-segments maythen be projected into two-dimensional value-spacewhere an efficient CG boundary seeking technique fin-ishes the job.

This work has been extended later first to rectangu-lar distance `1 in [13], then to partial coverage prob-lems in [12].A few years ago together with José Gordillo and

Emilio Carrizosa I studied [8] another similar push-pull location problem where the set R consists of ex-tensive facilities described as polygonal regions. Butinstead of looking at the bi-objective problem that hasa continuum of efficient solutions, we were looking forone particular solution that ‘best’ separates (if possi-ble) R from A. Separation was measured in a supportvector machine way (see e.g. [4]) by maximizing thedifference r2R−r2A where rR is the (euclidean) distanceto the closest r ∈ R and rA the farthest distance tosome a ∈ A. In the feasible case where A may beseparated by a circle from all regions in R, i.e. whenthis objective may be positive, this means geometri-cally that we seek the largest area annulus enclosingall points of A and not meeting R. In the non fea-sible case the objective will always be negative andwe look for the smallest area annulus that contains oroverlaps all A and such that its ‘hole’ does not meetR. These two cases are illustrated below; the annulusis coloured as light gray.

rA

rR

(Feasible case)

rA rR

(Unfeasible case)

For a site x we call ‘active’ any r closest to x (theactual closest point(s) of this r is also called active)and any a farthest from x . We showed that generi-cally three cases may arise at an optimal solution:

33

Three tapas with CG sauce

(1) there are 4 active elements with at least one ofeach type, or there are 3 active elements with one aand two r (in the nonconvex case possible twice thesame r with different active points) either (2) withcolinear active points, or (3) one active r active at avertex. Enumeration of all realizations of such casesleads to an O(n5) algorithm.However, these conditions are directly related to the

farthest point Voronoi diagram VA w.r.t. A and theclosest point Voronoi diagram VR w.r.t. the polygonsR. It is well known that edges of VR are parts ofbisectors between two polygons, and these consist ofsuccessive linear and parabolic pieces depending onwhether the active points of both polygons are of thesame type (a vertex or on an edge) or not. The pointswhere these pieces touch are called breakpoints.Now Case (1) may be realized in three ways: either

as a vertex of VA or as a vertex of VR or as the point ofintersection between an edge of VA and an edge of VR.Case (3) corresponds to a breakpoint of some edge ofVR and case (2) happens at a finite number of pointseasily constructed from VR.Therefore using the CG approach should lead to

much lower complexity. But this CG approach to theproblems remains to be done.It should be noted that in an optimal solution to

a multifacility version of this push-pull problem allsites will satisfy the same conditions, so the same setof candidate sites arises, but now several of such siteswill have to be combined.

References

[1] J. Brimberg, H.T. Kakhki, G.O.Wesolowsky,Location among regions with varying norms,Annals of Operations Research 122 (2003), 87–102.

[2] J. Brimberg, H. Kakhki, G. Wesolowsky, Locat-ing a single facility in the plane in the presenceof bounded regions and different norms, Journalof the Operational Research Society of Japan 48(2005) 135–147.

[3] S. Cheng, H. Na, A. Vigneron, Y. Wang, Ap-proximate Shortest Paths in Anisotropic Re-gions, SIAM Journal on Computing (2008) 38802–824.

[4] N. Cristianini and J. Shawe-Taylor. An Intro-duction to Support Vector Machines and OtherKernel-based Learning Methods. CambridgeUniversity Press, Cambridge, 2000.

[5] J. Fathali, M. Zaferanieh, Location problems inregions with lp and block norms, Iranian Jour-nal of Operations Research 2 (2011) 72–87.

[6] J. Fliege, Efficient dimension reduction inmultifacility location problems, Shaker Verlag,Aachen, 1997.

[7] L. Franco, F. Velasco, L. Gonzalez-Abril, Gatepoints in continuous location between regionswith different p norms, European Journal ofOperational Research 218 (2012) 648–655.

[8] J. Gordillo, F. Plastria, E. Carrizosa, Lo-cating a semi-obnoxious facility with re-pelling polygonal regions, working paper (2007)30p. http://www.optimization-online.org/DB_HTML/2007/04/1652.html

[9] L. Kaufman, F.Plastria, The Weber problemwith supply surplus, Belgian Journal of Oper-ations Research, Statistics and Computer Sci-ence 28 (1988) 15–31.

[10] J.S.B. Mitchell, C.H. Papadimitriou, Theweighted region problem: finding shortest pathsthrough a weighted planar subdivision, Journalof the ACM 38 (1991) 18–73.

[11] Y. Ohsawa, Bicriteria Euclidean location as-sociated with maximin and minimax criteria,Naval Research Logistics 47 (2000) 581–592

[12] Y. Ohsawa, F. Plastria, K. Tamura, Euclideanpush-pull partial covering problems, Computers& Operations Research 33 (2006) 3566–3582

[13] Y. Ohsawa, K. Tamura, Efficient location fora semi-obnoxious facility Annals of OperationsResearch 123 (2003) 173–188

[14] M. Parlar, Single facility location problemwith region-dependent distance metrics, Inter-national Journal of Systems Sciences 25 (1994)513–525.

[15] F.Plastria, Up and downgrading the euclidean1-median problem, in preparation (2013)

[16] F.Plastria, Shortest paths using varying dis-tance functions, in preparation (2013)

[17] F.Plastria, M.Elosmani, On the convergence ofthe Weiszfeld algorithm for continuous singlefacility location-allocation problems, TOP 16(2008) 388–406.

[18] F.Plastria, M.Elosmani, Continuous location ofan assembly station, TOP First online (April2011).

[19] M. Zaferanieh, H. Kakhki, J. Brimberg, G.Wesolowsky, A BSSS algorithm for the singlefacility location problem in two regions with dif-ferent norms, European Journal of OperationalResearch 190 (2008) 79–89.

34


On the barrier-resilience of arrangements of ray-sensors

David Kirkpatrick1, Boting Yang:2, and Sandra Zilles;2

1Department of Computer Science, University of British Columbia, Vancouver, Canada.2Department of Computer Science, University of Regina, Regina, Canada.

Abstract

Given an arrangement A of n sensors and twopoints s and t in the plane, the barrier resilienceof A with respect to s and t is the minimum num-ber of sensors whose removal permits a path froms to t such that the path does not intersect thecoverage region of any sensor in A. When thesurveillance domain is the entire plane and sen-sor coverage regions are unit line segments, evenwith restricted orientations, the problem of de-termining the barrier resilience is known to beNP-hard [11, 12]. On the other hand, if sensorcoverage regions are arbitrary lines, the problemhas a trivial linear time solution. In this paper,we give an Opn2mq time algorithm for computingthe barrier resilience when each sensor coverageregion is an arbitrary ray, where m is the numberof sensor intersections.

1 Introduction

Barrier coverage is an important coverage conceptthat arises in the analysis of wireless sensor net-works [3]. Other notions of coverage try to quan-tify the effectiveness of a collection of sensors indetecting the presence of objects in a particularsurveillance region. Barrier coverage, motivatedby applications such as border control, measuresthe effectiveness of detecting the movement of ob-jects between, but not necessarily within, criticalregions. Given a sensor network, specified as anarrangement A of sensors with associated cover-age regions in the plane, and two points s and t,we say that the sensor network provides barriercoverage if it guarantees that any object moving

Email: [email protected].:Email: [email protected].;Email: [email protected].

Research support for all three authors is provided by theNatural Sciences and Engineering Research Council ofCanada.

from point s to point t must be detected by atleast one sensor.

If sensor regions are connected then determin-ing barrier coverage amounts to asking if s and tbelong to the same face of the arrangement, whichis straightforward to check provided the sensor re-gion boundaries are sufficiently simple. In orderto measure robustness of barrier coverage, Kumaret al. [10] introduced k-barrier coverage that guar-antees that any path from a point s to a point tmust intersect at least k distinct sensor regions.They showed that for unit disk sensors (i.e., sen-sors whose coverage regions are unit disks) dis-tributed in a strip separating s and t, k-barriercoverage can be determined efficiently by reduc-tion to a maximum flow problem in the intersec-tion graph of the disks.

Bereg and Kirkpatrick [2] introduced and stud-ied the associated optimization problem, whichthey called barrier resilience. This specifies theminimum number of sensors whose removal per-mits a path from point s to point t such thatthe path does not penetrate any of the remainingsensor coverage regions. Bereg and Kirkpatrickshowed that there is a 3-approximation (or, un-der mild restrictions concerning the separation ofs and t, a 2-approximation) algorithm for com-puting the barrier resilience of unit disk sensors.When the sensor coverage regions are arbitraryline segments, Alt et al. [1] proved that the prob-lem of determining barrier resilience is NP-hardand APX-hard. In fact, even if all sensor coverageregions are unit line segments in one of at mosttwo orientations, the barrier resilience problem re-mains NP-hard [11, 12]. The reader is referred tothe papers [3,5,8] for more information on barriercoverage and related problems.

It is straightforward to see that if sensor cov-erage regions are arbitrary lines, the barrier re-silience problem has a linear time solution, sincethe resilience of a given arrangement of lines isjust the number of lines that separate s and t.

35

Resilience of ray-sensorsIn this paper, we address the case where sensorcoverage regions are half-lines (or rays). We de-scribe an Opn2mq time algorithm for computingthe resilience of an arbitrary arrangement of nrays with m intersections. (Due to space con-straints, proofs are omitted; full proofs will ap-pear in an expanded version of the paper.)

2 Ray barriers and barriergraphs

Let ~V be a set of n rays, specified by an endpointand a direction, in the plane. Suppose that weare given a sensor network consisting of n sensors,where the coverage regions of sensors correspondto the rays in ~V , and two points s and t which arenot intersected by any of the rays in ~V . We con-sider the problem of finding a subset ~U of rays in~V with the minimum cardinality such that thereis a path from s to t which does not intersect anyrays in ~V z~U . The cardinality of ~U is referred to asthe resilience of the sensor configuration ps, t, ~V q,and ~U is a resilience set of ps, t, ~V q.1We say that a set ~V 1 ~V forms a barrier for

ps, t, ~V q if any path from s to t intersects at leastone of the rays in ~V 1. Thus a set ~U ~V is a re-silience set of ps, t, ~V q if and only if ~U is a smallestsubset of ~V with the property that ~V z~U does notform a barrier. Our algorithm for computing aresilience set uses a reformulation of the problemas a graph problem. This reformulation is basedon the observation that if a set of rays forms abarrier it must contain a subset consisting of tworays that forms a barrier; we refer to such a subsetas a 2-barrier.In order to substantiate this observation, we

introduce some helpful notation. For two pointsa and b in the plane, we use ab to denote theline segment with endpoints a and b, and use ~a todenote a ray with endpoint a.For the remainder of this paper, suppose, with-

out loss of generality, that the line containing stis horizontal. (Accordingly, we will say “above(resp., below) st” as an abbreviation for “above(resp., below) the horizontal line supporting st”.)For any ray ~a P ~V , we assign a unique color as fol-lows: (i) if ~a intersects st and goes down we assignit the color red (drawn as a solid ray in figures);(ii) if ~a intersects st and goes up we assign it thecolor blue (drawn as a dashed ray in figures); and(iii) if ~a does not intersect st we assign it the color

1Note that a configuration ps, t, ~V q does not necessarilyhave a unique resilience set.

s t

(b)(a) (c)

s ts t

Figure 1: paq: a red-blue 2-barrier; pbq: a red-black 2-barrier; pcq: a blue-black 2-barrier

s t

(a) (b)

Figure 2: paq: a sensor configuration ps, t, ~V q; pbq:its associated barrier graph

black (drawn as a dotted ray in figures).We first observe that certain pairs of intersect-

ing rays are guaranteed to form a 2-barrier (seeFigure 1).

Proposition 1 Let ps, t, ~V q be a sensor config-uration. A pair of intersecting rays t~a,~bu ~Vforms a 2-barrier if (i) one is red and the otheris blue, (ii) one is red and the other is black andthey intersect above st, or (iii) one is blue and theother is black and they intersect below st.

Lemma 2 Let ps, t, ~V q be a sensor configurationand ~V 1 ~V . The set ~V 1 forms a barrier forps, t, ~V q if and only if there are two rays ~a,~b P ~V 1

such that t~a,~bu forms a 2-barrier of one of thetypes described in Proposition 1.

Lemma 2 motivates the reformulation of the re-silience problem as a graph problem (see Lemma 3below.) We say that a graph G pV,Eq is abarrier graph of ps, t, ~V q, denoted by Gp~V q, if V

is the set of all endpoints in ~V , and ta, bu P E

iff the corresponding pair of rays t~a,~bu forms a2-barrier (see Figure 2). It follows immediatelyfrom Lemma 2 that barrier graphs are tripartite.Note that we use the same notation for a ver-

tex in Gp~V q and an endpoint in ~V . When thereis no ambiguity, a vertex of Gp~V q is also re-ferred to as an endpoint of a ray. We exploit thevertex-endpoint duality to view Gp~V q as a vertex-coloured embedded graph. This allows us to say,for example, that any vertex that lies above stmust be red or black, and any vertex that liesbelow st must be blue or black.

36

XV Spanish Meeting on Computational Geometry, June 26-28, 2013Lemma 3 For any sensor configuration ps, t, ~V q,a vertex set Vc is a minimum size vertex cover ofGp~V q if and only if the corresponding set of rays~Vc is a resilience set of ps, t, ~V q.

From Lemma 3, it is clear that we can find aresilience set efficiently if there is an efficient al-gorithm to compute a minimum size vertex coverof the graph Gp~V q. Unfortunately, the vertexcover problem on general tripartite graphs is NP-complete (there is a straightforward reductionfrom the vertex cover problem for cubic planargraphs, which is known to be NP-complete [6]).In fact, Clementi et al. [4] have shown that theminimum size vertex cover for tripartite graphs isnot even approximable to within a factor of 3433,unless P=NP.Thus, we are motivated to look for additional

structure in instances of the tripartite vertex coverproblems that arise from barrier graphs. In somesettings, for example if all rays are parallel to oneof two different lines, the graph Gp~V q is easilyseen to be bipartite. It is well-known, by König’stheorem [9], that, for bipartite graphs, construct-ing a minimum size vertex cover is equivalent toconstructing a maximum size matching. Thus, wecan use the Hopcroft-Karp algorithm [7] to finda minimum size vertex cover in such instances inOpm?

nq time, where m is the number of edges inGp~V q .To exploit the structure inherent in less re-

stricted instances, we first introduce some addi-tional notation (see Figure 3). We denote by `sv

the line passing through points s and v. Simi-larly `tw denotes the line passing through pointst and w. A generic line through s (respectively,t) is denoted `s (respectively, `t). Similarly, adistinguished line through s (respectively, t) willbe denoted `s (respectively, `t). With any line`s (respectively, `t) we associate the half-space,denoted `s (respectively, `t) consisting of allpoints to the left of `s, or above `s in case `s

is horizontal (respectively, all points to the rightof `t, or above `t in case `t is horizontal).Armed with this notation, we can capture two

additional properties of barrier graphs that can beexploited in the efficient construction of optimalvertex covers:

Lemma 4 Let Gp~V q be a barrier graph and sup-pose that Vc is any vertex cover of Gp~V q. Thenthere must exist lines `s and `t, through s andt respectively, such that Vc contains all of the redand blue vertices that lie in `s Y `t.

ts

v

w

`tw`sv

`+tw`−sv

Figure 3: Lines (and associated half-spaces)through s and t.

Lemma 5 Let `s and `t be arbitrary linesthrough s and t respectively, and let RB denotethe set of red and blue vertices that lie in `sY`t.Then the subgraph of the barrier graph Gp~V q thatis induced by the vertices in V zRB, is bipartite.

3 Algorithm

In this section, we present an algorithm for com-puting resilience sets. Its correctness follows im-mediately from Lemma 3.

Algorithm 1: resilience.

Input: Sensor configuration ps, t, ~V q.Output: A resilience set for ps, t, ~V q.build the vertex-coloured barrier graph Gp~V q1

(described in Section 2)return min-vertex-coverpGp~V qq2

As we have already noted, a minimum size ver-tex cover of a bipartite graph can be found inpolynomial time [7]. So the basic idea of our al-gorithm is to exploit this by forming a sequenceof subsets U1, U2, . . . of V such that (i) for all i,G|pV zUiq, the subgraph of Gp~V q induced on thevertex set V zUi, is bipartite, and (ii) for some i,the minimum size vertex cover of G|pV zUiq, to-gether with the vertices in Ui, forms a minimumsize vertex cover of Gp~V q.We know, by Lemma 4, that for any minimum

size vertex cover Vc of Gp~V q there must exist lines`s and `t, through s and t respectively, suchthat Vc contains all of the vertices inRB, the setof red and blue vertices that lie in `s Y `t. Fur-thermore, the vertices of VczRB must be a min-imum size vertex cover of G|pV zRBq; otherwise

37

Resilience of ray-sensorsVc would not have minimum size. So our algo-rithm simply tries all possibilities for `s and `t,determines the associated set RB, finds a mini-mum size vertex cover of G|pV zRBq (which, byLemma 5, is bipartite), and chooses, among all ofthese possibilities, one that minimizes the size ofthis vertex cover together with the setRB.

Algorithm 2: min-vertex-cover of a bar-rier graph.

Input: A barrier graph Gp~V q.Output: A minimum vertex cover of Gp~V q.RB Ð tred vertices in V 1

V Ctemp Ð2

bipartite-vertex-coverpG|pV zRBqqV Cbest Ð V Ctemp YRB3

for every red vertex v do4

for every red vertex w do5RB Ð6

tred and blue vertices in `sv Y `twuV Ctemp Ð7

bipartite-vertex-coverpG|pV zRBqqif |V Ctemp| |RB| |V Cbest| then8

V Cbest Ð V Ctemp YRB9

return V Cbest10

As we have already noted, the correctness ofAlgorithm resilience follows immediately fromLemma 3. We now turn our attention to the cor-rectness of our vertex cover algorithm for bar-rier graphs.

Theorem 6 The output of Algorithm 2 is a min-imum size vertex cover of Gp~V q.It will be clear from the description of Algo-

rithm 2 that the problem of constructing a min-imum size vertex cover of a barrier graph with nvertices and m edges can be reduced to Opn2qminimum size vertex cover subproblems on in-duced subgraphs, each of which, by Lemma 5,is bipartite. As previously noted, König’s the-orem [9] states that, for bipartite graphs, con-structing a minimum size vertex cover is equiv-alent to constructing a maximum size matching.Thus, we can use the Hopcroft-Karp algorithmto find a minimum size vertex cover in each sub-problem in Opm?

nq time [7], or Opn2m?

nq timein total. We note, however, that it is possible toimplement Algorithm 2 to run in Opn2mq time,by ordering the successive subproblems in a waythat they do not require independent solution; we

leave the details to an expanded version of thispaper.

References[1] H. Alt, S. Cabello, P. Giannopoulos and C.

Knauer, Minimum cell connection and separationin line segment arrangements, arXiv:1104.4618v2[cs.CG], 2011.

[2] S. Bereg and D. Kirkpatrick, Approximating bar-rier resilience in wireless sensor networks, Pro-ceedings of the 5th International Workshop on Al-gorithmic Aspects of Wireless Sensor Networks,Lecture Notes in Computer Science, Vol. 5804,pages 29–40, 2009.

[3] M. Cardei and J. Wu, Coverage in wireless sen-sor networks, In M. Ilyas and I. Mahgoub, ed-itors, Handbook of Sensor Networks: CompactWireless and Wired Sensing Systems, chapter 19,pages 432–446, CRC Press, 2005.

[4] A. Clementi, P. Crescenzi, G. Rossi, On thecomplexity of approximating colored-graph prob-lems. Proceedings of the 5th International Con-ference on Computing and Combinatorics, Lec-ture Notes in Computer Science, Vol. 1627, pages281–290, 1999.

[5] A. Chen, T. H. Lai and D. Xuan, Measuring andguaranteeing quality of barrier-coverage in wire-less sensor networks, Proceedings of the 9th ACMinternational symposium on Mobile ad hoc net-working and computing, pages 421–430, 2008.

[6] M. Garey and D. Johnson, Computers and In-tractability: A Guide to the Theory of NP-Completeness, Freeman, San Francisco, 1979.

[7] J. E. Hopcroft and R. M. Karp, An n2.5 al-gorithm for maximum matchings in bipartitegraphs, SIAM Journal on Computing, 2(4): 225–231, 1973.

[8] S. Kloder and S. Hutchinson, Barrier coveragefor variable bounded-range line-ofsight guards,Proceedings of IEEE International Conference onRobotics and Automation (ICRA’07), pages 391–396, 2007.

[9] D. König, Gráfok és mátrixok, Matematikai ésFizikai Lapok, 38:116–119, 1931.

[10] S. Kumar, T. H. Lai, and A. Arora, Barrier cov-erage with wireless sensors, Wireless Networks,13(6): 817–834, 2007.

[11] K.-C. Tseng, Resilience of Wireless Sensor Net-works, Master thesis, University of BritishColumbia, BC, Canada, 2011.

[12] K.-C. Tseng and D. Kirkpatrick, On Barrier Re-silience of Sensor Networks, Proceedings of the7th International Workshop on Algorithmic As-pects of Wireless Sensor Networks, Lecture Notesin Computer Science, Vol. 7111, pages 130-144,2011.

38



Sergio Cabello∗1, Markus Chimani†2, and Petr Hliněný‡3

1Department of Mathematics, FMF, University of Ljubljana, Slovenia2Theoretical Computer Science, Department of Maths/CS, Osnabrück University, Germany

3Faculty of Informatics, Masaryk University, Brno, Czech Republic

Abstract

Let G be a graph embedded in an orientable surfaceΣ, possibly with edge weights, and denote by len(γ)the length (the number of edges or the sum of the edgeweights) of a cycle γ in G. The stretch of a graph em-bedded on a surface is the minimum of len(α) · len(β)over all pairs of cycles α and β that cross exactly once.We provide an algorithm to compute the stretch of anembedded graph in time O(g4n log n) with high prob-ability, or in time O(g4n log2 n) in the worst case,where g is the genus of the surface Σ and n is thenumber of vertices in G.

Introduction

Consider a graph G embedded on an orientable sur-face Σ of genus g. What can be said about the crossingnumber of G in the plane? Is it computable in poly-nomial time? If it is not, can we obtain a reasonableapproximation in polynomial time? Unfortunately,Cabello and Mohar [3] show that the crossing numberof such graphs is not computable in polynomial time,even when Σ is the torus. Djidjev and Vrt’o [4] showthat the crossing number of G is upper bounded byO(g∆n), where n is the number of vertices in G and ∆is the maximum degree of G. This is an improvementover the previous bound of O(Cg∆n), for some con-stant C, by Börözky, Pach and Tóth [1]. Under somemild assumptions about the density of the embeddingof G, Hliněný and Chimani [5] give a (3 · 23g+2∆2)-approximation algorithm for the crossing number ofG. This last work is the main motivation for our re-search.Hliněný and Chimani [5] define the stretch of an

∗Supported by the Slovenian Research Agency, programP1-0297, project J1-4106, and within the EUROCORES Pro-gramme EUROGIGA (project GReGAS) of the European Sci-ence Foundation.†This research was conducted while being funded by a Carl-

Zeiss-Foundation juniorprofessorship, and partially supportedby EUROCORES Programme EUROGIGA (project GraDR)of the European Science Foundation.‡Supported by the Czech Science Foundation, EURO-

CORES grant GIG/11/E023 (project GraDR).

embedded graph G as

str(G) = minα,βlen(α) · len(β),

where α, β ranges over all pairs of cycles in G thatcross exactly once. Here, len(α) denotes the num-ber of edges in α and a cycle is a closed walk in agraph without repeated vertices. A precise definitionof what “crossing exactly once" means is given in Sec-tion 1.2. The stretch plays a fundamental role in theiranalysis of the algorithm. The concept of stretch canbe generalized to the case of positive edge-weightedgraphs in a natural way: take len(α) to be the sumof the edge-weights along the cycle α. Henceforth wewill assume this more general definition of stretch.It is worth noting that, if two cycles α and β are

crossing once, then they must be both (surface-)non-separating. That is, cutting the surface along α or βdoes not disconnect the surface. This is so becauseany cycle crosses a (surface-)separating cycle an evennumber of times. Thus, when computing the stretch,we can restrict our attention to pairs (α, β) of non-separating cycles.In this paper we provide an algorithm to compute

the stretch of an embedded graph in time O(g4n log n)with high probability, or in time O(g4n log2 n) in theworst case, where g is the genus of the surface Σ andn is the number of vertices in G.

Overview of the approach. Let us provide an in-formal overview of the main ideas. We show the fol-lowing recursive property of the stretch: it is definedeither by the shortest non-separating cycle α∗ and oneother cycle crossing α∗ exactly once, or by the stretchof surface obtained by cutting along α∗ and pastingdisks. We do not prove this claim directly, but usinga detour through another concept: odd-stretch.The definition of odd-stretch resembles the defini-

tion of stretch, but we allow closed walks α and β,instead of just cycles, and allow an odd number ofcrossings, instead of exactly one crossing. It turnsout that the stretch and odd-stretch of a graph is thesame. However, working with the odd-stretch is eas-ier because we only need to take care of the parityof crossings and, when constructing new closed walks

39


via exchange arguments, we do not need to take careto construct cycles. Finally, we prove the aforemen-tioned recursive property for the odd-stretch factor.The eventual algorithm, given in Figure 1 is very

simple. However, there is a fine point we have totake care of to obtain a polynomial-time algorithm.Repeatedly cutting along shortest non-separating cy-cles and pasting disks may give rise to an exponentialgrowth in the size of the graphs: at each cut we makecopies of the vertices along the cycle and thus thenumber of vertices may nearly double at each itera-tion. However, if at some iteration we get a shortestnon-separating cycle with more vertices than the orig-inal graph, we can finish the recursive search. In thisway we avoid the potentially exponential growth inthe size of the graphs.

1 Odd-stretchIn this section we introduce the concept of odd-stretch, which is a generalization of stretch. We firstdiscuss crossings for curves in general position andthen crossings for closed walks in a graph. Finally, wedefine the odd-stretch, discuss some of its propertiesand, eventually, show that the odd-stretch is the sameas the stretch.

1.1 Crossings of curves in general po-sition

Two curves C and C ′ on a surface Σ are in generalposition if they have a finite number of intersectionsand, at each intersection, they cross transversally. Fortwo curves C = C(t) and C ′ = C ′(t) in general posi-tion, the set of crossings is

X(C,C ′) = x ∈ Σ | ∃t, t′ s.t. x = C(t) = C ′(t′).

Two curves C and C ′ in general position cross k timesif and only if k = |X(C,C ′)|. We will use the follow-ing (intuitive) fact: the number of crossings betweentwo closed curves, modulo 2, is invariant under smallperturbations of any of the two closed curves.

1.2 Crossings of closed walks

Two closed walks α and β in G cross k times if andonly if: there are arbitrarily small perturbations of αand β to general position that cross k times, and anysmall enough perturbation of α and β has at leastk intersecting points. Moreover, we can always as-sume that the crossings of the perturbations occur ina neighborhood of the vertices. We denote by cr(α, β)the number of crossings between α and β.For any closed walk α, the set of closed walks in

G that cross α an odd number of times satisfies theso-called 3-path condition. The next lemma states

this in an equivalent way for easier use later on. Thisproperty was already noted in [5].

Lemma 1 Let α be a closed walk and let γ be a closedwalk crossing α an odd number of times. Let x and ybe two vertices on γ and let π be some walk from x toy. Let γ′ be the closed walk defined by concatenatingγ[y → x] and π. Let γ′′ be the closed walk defined byconcatenating γ[x → y] and the reverse of π. Theneither γ′ or γ′′ cross α an odd number of times.

1.3 Odd-stretch of an embedded graphThe odd-stretch of an embedded graph G is

oddstr(G) = minα,βlen(α) · len(β),

where α, β ranges over all pairs of closed walks in Gthat cross an odd number of times. We next remarkthe two main differences with the previous stretch:α and β iterate over closed walks, instead of cycles,and the curves can cross an odd number of times,instead of exactly once. A priori, the stretch and theodd-stretch of an embedded graph are different. Aposteriori we can see that they are the same; this wasalready noted in [5].

Lemma 2 The odd-stretch of G and the stretch of Gare the same.

2 Algorithm for computing thestretch factor

We will use the following two properties:

Lemma 3 ([2]) Let G be a graph withm vertices em-bedded in a surface of genus g.

• We can compute a shortest non-separating cyclein time O(g2m logm) with high probability, or intime O(g2m log2m) in the worst case.

• For any given non-separating cycle α, we cancompute a shortest cycle of G that crosses α ex-actly once in time O(gm logm) with high proba-bility, or in time O(gm log2m) in the worst case.

Lemma 4 Let α be a shortest non-separating cyclein G. For any two vertices x and y on α, α containsa shortest path from x to y or from y and x.

The algorithm for computing the stretch of an em-bedded graph is given in Figure 1. We first discuss itstime complexity and then its correctness.

Lemma 5 Algorithm ComputeStretch has timecomplexity O(g4n log n) with high probability, orO(g4n log2 n) in the worst case, where n is the numberof vertices in G.

40


'

&

$

%

Algorithm ComputeStretchInput: graph G embedded in surface ΣOutput: stretch of G1. i← 1;2. (G1,Σ1)← (G,Σ);3. str ←∞;4. while Σi not the sphere and |V (Gi)| ≤ g · |V (G)| do5. αi ← shortest non-separating cycle in Gi;6. βi ← shortest cycle crossing αi exactly once;7. str ← minstr, len(αi) · len(βi);8. (Gi+1,Σi+1)← cut (Gi,Σi) along αi and attach disks to the boundaries;9. i← i+ 1;10. return str

Figure 1: Algorithm ComputeStretch to compute the stretch of an embedded graph.

Proof. Because of Lemma 3, in each iteration ofthe while loop we need O(g2i ni log ni) time whp, orO(g2i ni log2 ni) in the worst case, where ni is the num-ber of vertices in Gi and gi is the genus of Σi. Sinceni = O(gn) because of the condition for iterating thewhile loop and gi ≤ g, each iteration of the while looptakes O(g2(gn) log(gn)) = O(g3n log n) time whp, orO(g3n log2 n) time in the worst case. There are atmost g iterations of the while loop.

Lemma 6 Let α be a shortest non-separating cycleand let β be a shortest cycle crossing α exactly once.Let G′ be the embedded graph obtained from G bycutting along α and attaching a disk to the bound-aries. The stretch of G is the minimum betweenlen(α) · len(β) and the stretch of G′.

Proof. Let Σ be the surface where G is embeddedand let Σ′ be the surface where G′ is embedded. Sinceany two closed curves of G′ that cross an odd numberof times in Σ′ also cross an odd number of times in Σ,it is clear that

str(G) ≤ minstr(G′), len(α) · len(β).

Thus, we have to argue the other inequality. If (α, β)define the stretch of G, then the other inequality isobvious.Let us consider next the case where (α, β) do not

define the stretch of G; it holds that str(G) <len(α) · len(β). Let (γ∗, σ∗) be the pair of cycles thatdefine the stretch of G. If there are several such pairs,we choose one such that cr(γ∗, α)+cr(σ∗, α) is mini-mum. We distinguish 3 cases depending on the valuesof cr(γ∗, α) and cr(σ∗, α):

Case cr(γ∗, α) = cr(σ∗, α) = 0. In this case, γ∗ andσ∗ keep crossing once in Σ′, and thus str(G) =str(G′).

Case cr(γ∗, α) = 1 or cr(σ∗, α) = 1. This case can-not actually happen. Let us assume that

cr(γ∗, α) = 1; the other case is symmetric. Sinceγ∗ crosses α once and β is a shortest cycle cross-ing α once, we have len(β) ≤ len(γ∗). Using thatα is a shortest non-separating cycle we have

str(G) = len(γ∗) · len(σ∗) ≥ len(β) · len(α).

This implies that (α, β) actually define thestretch of G.

Case cr(γ∗, α) ≥ 2 or cr(σ∗, α) ≥ 2. This case can-not actually happen. Let us assume thatcr(γ∗, α) ≥ 2; the other case is symmetric. Letx and y be two crossings of γ∗ and α that areconsecutive along α. Because of Lemma 4, αcontains a shortest path between x and y. Letπ denote this shortest path. We can use π andγ∗ to construct two cycles γ′ and γ′′ that areshorter and cross α fewer times than γ∗. Becauseof Lemma 1, some γ ∈ γ′, γ′′ crosses σ∗ an oddnumber of times. The pair (γ, σ∗) contradicts thechoice of (γ∗, σ∗).

This finishes all cases. (Note that the second andthird cases are not mutually exclusive.)

Lemma 6 shows correctness of the algorithm if thecondition |V (Gi)| ≤ g · |V (G)| is true for each i =1, . . . , g. We next argue why we can finish the searchif at some iteration |V (Gi)| > g · |V (G)|.

Lemma 7 If, for some i, αi has more than |V (G)|vertices, then for any ` ≥ i

str(G) = minlen(αj) · len(βj) | j = 1, . . . , `− 1.

Proof. Assume, for the sake of this proof, that inthe algorithm ComputeStretch we drop testing thecondition |V (Gi)| ≤ g · |V (G)|. The algorithm thenmakes exactly g iterations and computes cycles αj , βjfor each j = 1, . . . , g. Because of Lemma 6 it holds

str(G) = minlen(αj) · len(βj) | j = 1, . . . , g.

41


αk

Σk+1 Σk

αk

Σk

α′k α′′k

vv′′v′

Wk+1 Wk+1 Wk Wk

W ′k

Figure 2: Figure for the proof of Lemma 7.

Assume that, at some iteration, the shortest non-separating cycle αi in Gi, has more than |V (G)| ver-tices. For each k < i, the cycle αi corresponds to aclosed walk Wk in the graph Gk. Moreover, the walkWk does not cross the cycle αk, for each k < i. Sinceαi has more than |V (G)| vertices, W1 repeats somevertex of G1 = G. This means that W1 is not a cycle.Let k be the maximum index, 1 ≤ k < i such that

Wk is not a cycle in Gk; thusWk+1 is a cycle in Gk+1.Since W1 is not a cycle and Wi is a cycle, the indexk is well defined. See Figure 2, left and center. Let vbe a vertex of Gk that is repeated in Wk. Cutting Gkthrough αk produces two copies α′k and α′′k of αk. Letv′ and v′′ be the corresponding copies of v. We canform a closed walk W ′k in Gk by taking the subwalkof Wk from the first appearance of v until the sec-ond. This closed walk W ′k crosses αk once. Thereforelen(βk) ≤ len(W ′k) < len(Wk) = len(αi) ≤ len(αj) foreach j ≥ i. We conclude that, for each j ≥ i,

len(αj)·len(βj) > len(βk)·len(αj) ≥ len(βk)·len(αk).

Since k < i ≤ ` we conclude that

str(G) = minlen(αj) · len(βj) | j = 1, . . . , g= minlen(αj) · len(βj) | j = 1, . . . , `− 1.

Theorem 8 Let G be a graph with n vertices embed-ded in surface of genus g. We can compute the stretchof G in time O(g4n log n) with high probability, or intime O(g4n log2 n) in the worst case.

Proof. The time bound follows from Lemma 5. Forthe correctness, note that the while loop has the fol-lowing invariant: at the start of iteration i, the vari-able str stores the value

min(len(αj) · len(βj) | j = 1, . . . , i− 1 ∪ ∞).

If |V (Gi)| ≤ g · |V (G)| for each iteration, then thealgorithm finishes with i = g + 1, the surface Σg+1 is

a sphere, and correctness follows from Lemma 6. Ifat some iteration ` we have |V (G`)| > g · |V (G)|, thenthere was some index i < ` such that the cycle αi hadmore than |V (G)| vertices. Correctness then followsfrom Lemma 7.

Acknowledgments. We are grateful to DanielŠtefankovič for some early discussions.

References[1] K. J. Börözky, J. Pach, and G. Tóth. Planar

crossing numbers of graphs embeddable in anothersurface. Int. J. Found. Comput. Sci., 17(5):1005–1016, 2006.

[2] S. Cabello, E. W. Chambers, and J. Erick-son. Multiple-source shortest paths in embed-ded graphs, 2013. Accepted to SIAM J. Com-puting. Preprint available at http://arxiv.org/abs/1202.0314. Preliminary version at SODA2007.

[3] S. Cabello and B. Mohar. Adding one edge toplanar graphs makes crossing number hard. InProc. SoCG 2010, pages 68–76, 2010. See http://arxiv.org/abs/1203.5944 for the full version.

[4] H. Djidjev and I. Vrt’o. Planar crossing numbersof graphs of bounded genus. Discrete & Compu-tational Geometry, 48(2):393–415, 2012.

[5] P. Hliněný and M. Chimani. Approximating thecrossing number of graphs embeddable in any ori-entable surface. In Proc. SODA 2010, pages 918–927, 2010.

42


An algorithm that constructs irreducible triangulations ofonce-punctured surfaces

M. J. Chávez∗1, S. Lawrencenko†2, J. R. Portillo‡1, and M. T. Villar§1

1Universidad de Sevilla, Spain2Russian State University of Tourism and Service, Lyubertsy, Moscow Region, Russia

AbstractA triangulation of a surface is irreducible if there isno edge whose contraction produces another triangu-lation of the surface. In this work we propose an al-gorithm that constructs the set of irreducible trian-gulations of any surface with precisely one boundarycomponent.

Introduction and terminologyWe restrict our attention to the following objects:

• S = Sg orNk is the closed orientable surface Sg ofgenus g or closed nonorientable surface of nonori-entable genus k. In particular, S0 and S1 are thesphere and torus, N1 and N2 are the projectiveplane and the Klein bottle (respectively).

• S − D is S minus an open disk D (the hole).This compact surface is called the once-puncturedsurface. We assume that the boundary ∂S of S,which is equal to ∂D, is homeomorphic to a circle.

We use the notation Σ whenever we assume the gen-eral case - that is, Σ ∈ S, S −D.If a (finite, undirected, simple) graph G is 2-cell

embedded in Σ, the components of Σ − G are calledfaces. A triangulation of Σ with graph G is a 2-cellembedding T : G→ Σ in which each face is boundedby a 3-cycle (that is, a cycle of length 3) of G and anytwo faces are either disjoint, share a single vertex,or share a single edge. We denote by V = V (T ),E = E(T ), and F = F (T ) the sets of the vertices, theedges, and the faces of T , respectively. Equivalently,the triangulation T of a surface can be defined as ahypergraph of rank 3 or a 3-graph, with the vertexset V (T ) and the collection F (T ) of triplets of V (T )(called 3-edges, or triangles, of T ) (see [4]).By G(T ) we denote the graph (V (T ), E(T )) of tri-

angulation T . Two triangulations T1 and T2 are called∗Email: [email protected]. Research supported by PAI FQM-

189†Email: [email protected].‡Email: [email protected]. Research supported by PAI FQM-164§Email: [email protected]. Research supported by PAI FQM-164

isomorphic, denoted T1 ∼= T2, if there is a bijectionα : V (T1)→ V (T2) such that uvw ∈ F (T1) if and onlyif α(u)α(v)α(w) ∈ F (T2). Throughout this paper wedistinguish triangulations only up to isomorphism.In the case Σ = S − D let ∂T , which is equal to

∂D, denote the boundary cycle of T . The vertices andedges of ∂T are called boundary vertices and boundaryedges of T , respectively.A triangulation of a punctured surface is irreducible

(term which is more accurately defined in Section 1)if no edge can be shrunk without producing multipleedges or changing the topological type of the surface.The irreducible triangulations of Σ form a basis forthe family of all triangulations of Σ, in the sense thatany triangulation of Σ can be obtained from a memberof the basis by applying the splitting operation (intro-duced in Section 1) a finite number of times. To havesuch a basis in hand can be very useful in practicalapplication of triangulation generating; the papers [6]and [19] are worth mentioning.It is known that for every surface Σ the basis of

irreducible triangulations is finite (the case of closedsurfaces is proved in [2], [13], [12], and [7] and thecase of surfaces with boundary is proved in [3]). Atpresent such bases are known only for seven closedsurfaces and two once-punctured surfaces: the sphere([14]), projective plane ([1]), torus ([8]), Klein bottle([9] and [15]), S2, N3, and N4 ([16, 17]), the disk andMöbius band ([5]).In this paper we present an algorithm which is

designed as an application of some recent advanceson the study of irreducible triangulations of once-punctured surface collected in [5]. Specifically, Lem-mas 1-3 (in Section 1) are the main supporting the-oretical results for the mentioned algorithm. As aparticular example, all the non-isomorphic combina-torial types (293 in number) of triangulations of theonce-punctured torus are determined.

1 Previous results

Let T be a triangulation of Σ. An unordered pairof distinct adjacent edges vu and vw of T is called

43


a corner of T at vertex v, denoted by 〈u, v, w〉(=〈w, v, u〉). The splitting of a corner 〈u, v, w〉, de-noted by sp〈u, v, w〉, is the operation which consistsof cutting T open along the edges vu and vw andthen closing the resulting hole with two new triangu-lar faces, v′v′′u and v′v′′w, where v′ and v′′ denotethe two images of vertex v appearing as a result ofcutting. Under this operation, vertex v is extendedto the edge v′v′′ and the two faces having this edge incommon are inserted into the triangulation; thereforethe order increases by one and the number of edgesincreases by three.If a corner 〈u, v, w〉 is composed of two edges vu

and vw neighboring around vertex v, sp〈u, v, w〉 isequivalent to the stellar subdivision of the face uvw.Especially in the case Σ = S − D, uv ∈

E(T ), and v ∈ V (T ), the operation sp〈u, v] of split-ting a truncated corner 〈u, v] produces a single trian-gular face uv′v′′, where v′v′′ ∈ E(∂(sp〈u, v](T ))).Under the inverse operation, shrinking the edge

v′v′′, denoted by sh〉v′v′′〈, this edge collapses to asingle vertex v, the faces v′v′′u and v′v′′w collapseto the edges vu and vw, respectively. Thereforesp〈u, v, w〉 sh〉v′v′′〈(T ) = T . It should be noticedthat in the case Σ = S − D, v′v′′ ∈ E(∂T ), thereis only one face incident with v′v′′, and only that sin-gle face collapses to an edge under sh〉v′v′′〈. Clearly,the operation of splitting doesn’t change the topolog-ical type of Σ if Σ ∈ S, S − D. We demand thatthe shrinking operation must preserve the topologicaltype of Σ as well; moreover, multiple edges must notbe created in a triangulation. A 3-cycle of T is callednonfacial if it doesn’t bound a face of T . In the casein which an edge e ∈ E(T ) occurs in some nonfacial3-cycle, if we still insist on shrinking e, multiple edgeswould be produced, which would expel sh〉e〈(T ) fromthe class of triangulations. An edge e is called shrink-able or a cable if sh〉e〈(T ) is still a triangulation ofΣ; otherwise the edge is called unshrinkable or a rod.The subgraph of G(T ) made up of all cables is calledthe cable-subgraph of G(T ).The impediments to edge shrinkability in a trian-

gulation T of a punctured surface S−D are identifiedin [2, 3, 1, 8]; an edge e ∈ E(T ) is a rod if and only ife satisfies one of the following conditions:(1) e is in a nonfacial 3-cycle of G(T ). In particular,

e is a boundary edge in the case in which the boundarycycle is a 3-cycle.(2) e is a chord of D -that is, the end vertices of e

are in V (∂D) but e /∈ E(∂D).From now on, we assume that S 6= S0 and make

an agreement that by “non-facial 3-cycle” we mean anon-null-homotopic 3-cycle whenever we refer to con-ditions (1) and (2). Therefore, an edge e is a rod pro-vided e occurs in some non-null-homotopic 3-cycle,and e is a cable otherwise. Especially, the boundaryedges of stellar subdivided faces are now regarded as

cables unless they occur in some non-null-homotopic3-cycles. The convenience of this agreement is thatonce a rod turns into a cable in the course of anysplitting sequence, it always remains a cable underforthcoming splittings.A triangulation is said to be irreducible if it is free

of cables or equivalently, each edge is a rod. For in-stance, a single triangle is the only irreducible trian-gulation of the disk S0 −D.Let T be an irreducible triangulation of a punctured

surface S −D where S 6= S0. Let us close the hole inT by restoring the disk D add a vertex, p, in D, andjoin p to the vertices in ∂D. We thus obtain a triangu-lation, T ∗, of the closed surface S. In this setting wecall D the patch, call p the central vertex of the patch,and say that T is obtained from the correspondingtriangulation T ∗ of S by the patch removal.Notice that T ∗ may be an irreducible triangulation

of S but not necessarily. Using the assumption thatT is irreducible and the fact that each cable of T ∗fails to satisfy condition (1) (in the strong non-null-homotopic sense), it can be easily seen that in thecase T ∗ is not irreducible, all cables of T ∗ have to beentirely in D ∪ ∂D and, moreover, there is no cablewhich is entirely in ∂D whenever the length of theboundary cycle ∂D is greater than or equal to 4. Inparticular, we observe that each chord of D (if any) isa rod in T because it meets condition (2), and is alsoa rod in T because it meets condition (1). We nowcome to a lemma which is to be stated shortly aftersome necessary definitions.A vertex of a triangulation R of S is called a pylonic

vertex if that vertex is incident with all cables of R. Atriangulation which has at least one cable and at leastone pylonic vertex is called a pylonic triangulation.A triangulation may have a unique cable and there-

fore two pylonic vertices. However, if the number ofcables in a pylonic triangulation R is at least 2, R hasexactly one pylonic vertex.

Lemma 1 Suppose an irreducible triangulation T ofa punctured surface S −D (S 6= S0) is obtained fromthe corresponding triangulation T ∗ of S by the patchremoval. If T ∗ has at least two cables, then either thecentral vertex p of the patch is the only pylonic vertexof T ∗, or else the length of ∂D is equal to 3.

Let Ξn = Ξn(S) denote the set of triangulations ofa fixed closed surface S that can be obtained from anirreducible triangulation of S by a sequence of exactlyn repeated splittings.In the following result, by the “removal of a vertex

v” we mean the removal of v together with the inte-riors of the edges and faces incident with v and bythe “removal of a face” we mean the removal of theinterior of that face.

44


Lemma 2 Each irreducible triangulation T of S−D(S 6= S0) can be obtained either

(i) by removing a vertex from a triangulation inΞ0 = Ξ0(S), or

(ii) by removing a pylonic vertex from a triangulationin Ξ1 ∪ Ξ2 ∪ · · · ∪ ΞK , where the constant K isprovided by [3] (whenever a pylonic triangulationoccurs), or

(iii) by removing either of the two faces containinga cable in their boundary 3-cycles provided thatcable is unique in a triangulation in Ξ1 (wheneversuch a situation occurs), or

(iv) by removing the face containing two, or three, ca-bles in its boundary 3-cycle provided those two,or three, cables collectively form the whole cable-subgraph in a triangulation in Ξ1 ∪ Ξ2 (if such asituation occurs).

Lemma 3 If a triangulation of S has at least two ca-bles but has no pylonic vertex, then no pylonic vertexcan be created under further splitting of the triangu-lation.

2 Sketch of the algorithm

In this section triangulations are considered to behypergraphs of rank 3 or 3-graphs. Let T be a 3-graph with V = V (T ) and F = F (T ) the setsof the vertices and the triangles of T . This 3-graph can be uniquely represented as a bipartitegraph BT = (V (BT ), E(BT )) in the following way:V (BT ) = V (T )∪F (T ), uv ∈ E(BT ) if and only if thevertex u lies in the triangle v in T .The algorithm input is the set I of irreducible tri-

angulations of a closed surface S 6= S0. The outputof the algorithm is the set of all non-isomorphic com-binatorial types of irreducible triangulations of theonce-punctured surface S −D.The first step is the generation of the set Ξ1 ∪ Ξ2

from the set I. Next, every 3-graph T ∈ Ξ1 ∪ Ξ2

is then represented by their corresponding bipartitegraph BT . This has been implemented with the com-putational package Mathematica ([18]).The second step consists of discarding all duplicate

bipartite graphs and then, all duplicate triangulationsin Ξ1∪Ξ2 will be discarded. That is, the obtention ofall non-isomorphic triangulations, denoted Ξ1∪ Ξ2 re-spectively, which is implemented with the computingpackages Nauty (and gtools, [10], [11]).Next, all pylonic vertices in Ξ1∪Ξ2 are detected and

operations (i)–(iv) described in Lemma 2 are appliedto obtain irreducible triangulations (using Mathemat-ica).

If Ξ2 has no pylonic triangulation, immediately pro-ceed to the final step: Discard all duplicate triangula-tions by using Nauty and gtools. Otherwise generateΞ3 and apply the preceding steps to Ξ3. Repeat thisprocedure to process Ξ4,Ξ5, . . . until no pylonic tri-angulation is left in the current Ξn; then the processterminates and a required output is produced.The validity of this procedure is justified by Lem-

mas 1 - 3 along with the results of [3]. In particular,the finiteness of the procedure is deduced from theupper bound [3] on the number of vertices in an irre-ducible triangulation of S−D. In particular, that up-per bound implies (along with Lemma 3) that Ξn doesnot have a pylonic triangulation for any n ≥ K + 1,where K = 945 for S1 and K = 376 for N1. In realityK is much less than these values. By computer ver-ification (and also by hand) we have checked that infact K = 1 for S1, and K = 2 for N1.Let us now mention two examples.Firstly, this algorithm has been implemented for the

set of two irreducible triangulations of N1 ([1]). Thealgorithm gives a set of 6 irreducible triangulations ofthe Möbius band, N1−D, which is precisely the sameas that obtained by some of the authors of this workin [5], although by using no computational package.Secondly, we introduce the details of the torus case,

S1.

Example: the once-punctured torus

Input: The set of 21 irreducible triangulations of S1

(as they are labelled in [8]).

• Ξ1(S1) has 433 non-isomorphic triangulations:232 of them have no pylonic vertex, 193 havean only pylonic vertex and 8 have two pylonicvertices.

• Ξ2(S1) has 11612 non-isomorphic triangulations,none of them is a pylonic triangulation.

• Operations described in Lemma 2 provide:

(i) 184 triangulations; only 80 of them are non-isomorphic.

(ii) 209 triangulations; only 203 of them arenon-isomorphic.

(iii) 16 triangulations, 10 of them are non-isomorphic.

(iv) 0 triangulations.

Output: 293 non-isomorphic combinatorial types ofirreducible triangulations of the once-punctured torusS1 −D.

3 Final conclusionIt is clear that this algorithm can be implemented forany closed surface whenever its basis of irreducible

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triangulations is known. In a future contribution wehope to present the set of irreducible triangulations ofthe once-punctured Klein bottle, N2 −D.

References[1] D. Barnette (D. W. Barnette). Generating the trian-

gulations of the projective plane. J. Comb. Theory,Ser. B, 33, (1982), 222–230.

[2] D. W. Barnette, A. L. Edelson. All 2-manifolds havefinitely many minimal triangulations. Isr. J. Math.,67, (1989), No. 1:123–128.

[3] A. Boulch, É. Colin de Verdière, A. Nakamoto. Ir-reducible triangulations of surfaces with boundary.Graphs and Combinatorics DOI 10.1007/s00373-012-1244-1, 2012.

[4] J. Bracho, R. Strausz, Nonisomorphic complete tri-angulations of a surface. Discrete Math. 232, (2001),11–18.

[5] M. J. Chávez, S. A. Lawrencenko, A. Quintero, M. T.Villar. Irreducible triangulations of the Möbius band.Preprint 2013.

[6] L. Giomi, M. J. Bowick. Elastic theory of defects intoroidal crystals. Eur. Phys. J. E., 27, (2008), 275–296.

[7] G. Joret, D. R. Wood. Irreducible triangulations aresmall. J. Comb. Theory, Ser. B, 100, (2010), No.5:446–455.

[8] S. A. Lavrenchenko (S. Lawrencenko). Irreducible tri-angulations of the torus. J. Sov. Math., 51, No. 5(1990), 2537–2543; translation from Ukr. Geom. Sb.30, (1987), 52–62.

[9] S. Lawrencenko, S. Negami. Irreducible triangula-tions of the Klein bottle. J. Comb. Theory, Ser. B,70, No. 2 (1997), 265–291.

[10] B. McKay, Practical Graph Isomorphism. CongressusNumerantium, 30, (1981), 45–87.

[11] B. McKay, nauty User’s Guide (Version 2.4)http://pallini.di.uniroma1.it/

[12] A. Nakamoto, K. Ota. Note on irreducible triangu-lations of surfaces. J. Graph Theory, 20, (1995), No.2:227–233.

[13] S. Negami. Diagonal flips in pseudo-triangulations onclosed surfaces. Discrete Math., 240, (2001), No. 1–3:187–196. 2001.

[14] E. Steinitz, H. Rademacher, Vorlesungen über dieTheorie der Polyeder unter Einschluss der Elementeder Topologie. Berlin: Springer, 1976. [Reprint of theoriginal 1934 edition.]

[15] T. Sulanke. Note on the irreducible triangulations ofthe Klein bottle. J. Comb. Theory, Ser. B, 96, No.6, (2006), 964–972.

[16] T. Sulanke. Generating irreducible triangulationsof surfaces, preprint, 2006, arXiv:math/0606687v1[math.CO].

[17] T. Sulanke. Irreducible triangulations of low genussurfaces, preprint, 2006, arXiv:math/0606690v1[math.CO].

[18] Wolfram Research, Inc. Mathematica Version 8.0(Wolfram Research, Inc.) Champaign, Illinois 2010.

[19] Y. Zhongwey, J. Shouwei. STL file generation fromdigitised data points based on triangulation of 3Dparametric surfaces.Int. J. Adv. Manuf. Technol. 23,(2004), 882–888.

46



Ana Paula Tomás∗

DCC & CMUP, Faculdade de CiênciasUniversidade do Porto, Portugal

Abstract

Although the exact counting and enumeration of poly-ominoes remain challenging open problems, severalpositive results were achieved for special classes ofpolyominoes. We give an algorithm for direct enumer-ation of permutominoes [12] by size, or, equivalently,for the enumeration of grid orthogonal polygons [19].We show how the construction technique allows us toderive a simple characterization of the class of convexpermutominoes, which has been extensively investi-gated recently [4]. The approach extends to some ofits subclasses, namely to the row convex and the di-rected convex permutominoes.

Introduction

The generation of geometric objects has applicationsto the experimental evaluation and testing of geo-metric algorithms. No polynomial time algorithmis known for generating polygons uniformly on agiven set of vertices. Some generators employ heuris-tics [1, 6] or restrict to certain classes of polygons, e.g.,monotone, convex or star-shaped polygons [18, 20].Numerous related problems have also been extensivelyinvestigated, as the exact counting or enumerationof polyominoes [9]. These remain challenging openproblems in computational geometry and enumera-tive combinatorics. A polyomino is an edge-connectedset of unit squares on a regular square lattice (grid).Polyominoes are dened up to translations. In thispaper, we give an algorithm for the enumeration ofpermutominoes by size, or, equivalently, for the enu-meration of grid orthogonal polygons (see Fig. 1).Polyominoes are usually enumerated by area (i.e.,number of cells). The direct enumeration of polyomi-noes is a computational problem of exponential com-plexity. An overview of the main developments con-cerning direct and indirect approaches is given in [3].Jensen's transfer-matrix algorithm [13] an indirect

∗Email: [email protected]. Research partially supportedby the European Regional Development Fund through theprogramme COMPETE and by the Portuguese Governmentthrough the FCT Fundação para a Ciência e Tecnologia un-der the project PEst-C/MAT/UI0144/2011, and project JEDI(PTDC/EIA/66924/2006).

Figure 1: Permutominoes of size 1, 2, and 3 (i.e., with4, 6, and 8 vertices) and their horizontal partitions.

method is currently the most powerful algorithm forcounting xed polyominoes. Exact counts are knownfor polyominoes that have up to 56 cells [3, 14]. Asfar as we can see, Jensen's algorithm cannot be easilyadapted for counting permutominoes.

1 Background

A polygon is called orthogonal if its edges meet atright angles. If r is the number of reex vertices ofan n-vertex orthogonal polygon, then n = 2r + 4(e.g. [16]). Grid orthogonal polygons (grid ogons)were introduced in [19] as a relevant class for gener-ation. A grid ogon is an orthogonal polygon withoutcollinear edges, embedded in a regular square grid,and having exactly one edge in each line of its mini-mal bounding square. They were addressed more re-cently under the name of permutominoes [4, 8, 12],because they correspond to polyominoes that are de-termined by a suitable pair of permutations havingthe same size. All polyominoes we will consider aresimply-connected and, similarly, all polygons are sim-ple and without holes. A permutomino that is givenby permutations of 1, 2, . . . , r+ 2, for r ≥ 0, is saidto have size r + 1. The size is the width of its min-imal bounding square. The topological border of apermutomino of size r+ 1 is a grid ogon with r reexvertices, and so, it has n = 2r + 4 vertices in total.In this paper, we give an algorithm for the enu-

meration of all permutominoes by size, based on theInflate-Paste

1 technique introduced in [19]. Every

1Demos at http://www.dcc.fc.up.pt/apt/genpoly

47


grid n-ogon P results from a unit square by apply-ing Inflate-Paste r times, where r = (n − 4)/2 isthe number of reex vertices of P . Inflate-Paste

glues a new rectangle to a grid ogon to obtain a newone with 1 more reex vertex. The rectangle is gluedby Paste to an horizontal edge incident to a convexvertex, say v, must be contained in a region that wecalled the free neighbourhood of v (Fig. 2), and is xedto v. This region, denoted by FSN(v), consists of the

Figure 2: Inflate-Paste: (a) gluing a rectangle to v(b) FSN(v) is the shaded region (c) the rectangle isdened by v and the center of a cell of FSN(v).

external points that are rectangularly visible from vand belong to the quadrant with origin v that con-tains the horizontal edge incident to v (two points aand b are rectangularly visible if the axis-aligned rect-angle that has a and b as opposite corners does notintersect the interior of the polygon). The Inflate

operation keeps the grid regular: the grid lines areshifted, if needed, to insert two new horizontal andvertical lines for the new edges (Fig. 3).

Figure 3: Two applications of Inflate-Paste: thecenter of the inated cell becomes a convex vertex ofthe polygon created at each step.

In [19], FSN(v) was called the free staircase neigh-borhood of v. Actually, FSN(v) is a Ferrers diagram,with origin at v, and hence it can be given by a se-quence of integers, each integer representing the num-ber of cells that form each row of the diagram. Anycell in FSN(v) could be used for growing the polygonusing v. Therefore, for the example given in Fig. 2,we could produce 9+7+6+4+4+3+2 = 35 distinctgrid ogons using the selected vertex.

2 Direct Enumeration

ECO was introduced in [2] as a construction paradigmfor the enumeration of combinatorial objects of a givenclass, by performing local transformations that in-crease a certain parameter (the size) of the objects.

In this section we propose a direct enumeration pro-cedure for grid ogons (i.e., permutominoes) usingInflate-Paste. It is based on the existence of aunique depth-rst generating tree for each n-ogon,once we x the order for visiting the horizontal par-tition. One possibility is to dene it as the order in-duced by a clockwise walk around the polygon, start-ing at its southwest vertex, as in Fig. 4. The bottom

Figure 4: The unique generating tree for the repre-sented grid ogon. Vertices 1, 2, 3 and 4 in the polygonare the ones that could still be expanded in our enu-meration method, and should be visited in this order.

line (i.e., the horizontal line that contains the SW-vertex) is never moved upwards, but polygons canmove freely along it. Fig. 1 shows how permutomi-noes can be generated by our method. The verticesmarked with a cross would be no longer available forexpansion.

PermutominoEnum(P ,S,G,n0,r)if r = 0 then return

MakeEmpty(TrS)while not IsEmpty(S) do

v := Pop(S) /* v is (vx, vy) */eH(v) := the horizontal edge of P that contains vif IsConvex(v,P ) then

for C in FreeNeighbourhood(v, P, G) do(p, q) := the southwest corner of CInflateGrid(p,q,G)w1 := (p + 1,vy); w2 := (p + 1,q + 1);w3 := (vx,q + 1)PasteRectangle(v, [w1, w2, w3], P )if w1 ∈ Interior(eH(v)) then

Push(w2, w3,S,P )else Push(w1, w2, w3,S,P ) OutputPolygon(P ,n0 + 2)PermutominoEnum(P ,S,G,n0 + 2,r − 1)CutRectangle(v, [w1, w2, w3], P )DeflateGrid(p + 1,q + 1,G)

done

Push(v,TrS);done

while not IsEmpty(TrS) doPush(Pop(TrS),S)

done

Here, P is the initial polygon, G a representationof the grid lines, S a stack that contains the con-vex vertices of P that are available for expansion,n0 the number of vertices of P and r the maximumnumber of reex vertices of the polygons. Permu-

tominoEnum enumerates recursively all descendants

48


of P that have up to n0 + 2r vertices. If initiallyP := (1, 1), (2, 1), (2, 2), (1, 2) (w.r.t. the standard2D cartesian coordinate system and given in CCW-order), S := (1, 2), (2, 2), and n0 = 4, then the al-gorithm will enumerate and count all grid ogons thathave up to 2r + 4 vertices.

In contrast to other existing methods for the enu-meration of polyominoes, PermutominoEnum, forpermutominoes, does not need to keep an exponen-tial number of state congurations in order to countthem correctly. Each permutomino is generated ex-actly once and, hence, there is no need to check forrepetitions. Nevertheless, the running time of the al-gorithm is dominated by the number of permutomi-noes generated (and thus it is exponential).

An algorithm for enumerating the convex permu-tominoes by size was published recently [10]. Its run-ning cost is proportional to the number of permutomi-noes generated. It is quite easy to design a specializedversion of our algorithm for enumerating convex per-mutominoes with identical complexity. Actually, aswe will see, for convex permutominoes the free neigh-bourhoods are linear (rectangles of width 1) and onlythe two topmost convex vertices can be active.

3 Convex Permutominoes

Although the exact counting and enumeration of poly-ominoes remain challenging open problems, severalpositive results were achieved for special classes ofpolyominoes [5, 7, 15], namely for the class of con-vex polyominoes and some of its subfamilies (e.g.,directed-convex polyominoes, parallelogram polyomi-noes, stack polyominoes, and Ferrers diagrams). Thelarger class of row-convex (resp. column-convex) poly-ominoes was considered also [11]. A polyomino is saidto be row-convex (resp. column-convex) if all its rows(resp. columns) are connected, i.e., the associated or-thogonal polygon is y-monotone (x-monotone). Apolyomino is said to be convex if it is both row-convexand column-convex. These classes, which satisfy con-vexity and/or directness conditions, have been studiedusing dierent approaches and are fairly well charac-terized, for some parameters, e.g., area and perime-ter [5]. The corresponding classes of permutominoeshave been addressed recently [4, 8].

The analysis of the transformations performed byInflate-Paste during the application of Permu-

tominoEnum allow us to derive simple characteri-zations and exact countings for such classes of per-mutominoes. Fig. 5 shows all n-vertex convex per-mutominoes for n = 4, 6, 8, each one embedded on agrid. Only the two topmost convex vertices may beactive for Inflate-Paste (so, L and R stand for leftor right). Crossed vertices are inactive in the follow-ing transformation steps: u means that the vertex

Figure 5: Enumerating convex permutominoes bysize.

would be discarded in PermutominoEnum as well(due to uniqueness conditions) and c means thatthe resulting permutomino would not be convex. Thesequence of 0, 1, 2? displayed on the grid top row isthe expansion key of the corresponding permutomino.Each element of the key gives the number of activeconvex vertices that see a certain grid cell (in Fig. 5,each counter is in its cell). Here, see means that thecell belongs to the free neighborhood of the vertex.For all the remaining empty cells, the counter is 0and, thus, we omitted it. If we add up the elementsof the expansion key of a given convex permutomino,we get the number of convex permutominoes that ityields immediately in PermutaminoEnum. In thisway, the expansion keys provide an exact encodingof the structural features that are relevant for count-ing convex permutominoes according to the numberof vertices. Actually, it is the key as a whole thatmatters but not the particular cells associated to eachcounter. By analysing Inflate-Paste in the scopeof PermutaminoEnum, we may conclude that theexpansion key of any convex permutomino with r ≥ 0reex vertices must be of one of the following forms:

1 2r+1 11 2j 0, for 1 ≤ j ≤ r−1,0 2j 1, for 1 ≤ j ≤ r−1, and0 2j 0, for 1 ≤ j ≤ r−2.

Inflate-Paste operations acting on convex permu-tominoes can be seen as rewrite rules. Each rulerewrites the key of a convex permutomino with r − 1reex vertices to the key of one of the convex per-mutominoes derived from it, having one more reexvertex, for r ≥ 1. The rewrite rules are:

12r1 →L,R 12r+1112r1 →L 12j0, for 1 ≤ j ≤ r12r1 →R 02j1, for 1 ≤ j ≤ r12j

′0 →L 12j0, for 1 ≤ j ≤ j′ ≤ r−1

12j′0 →R 12j

′+10, for 1 ≤ j′ ≤ r−112j

′0 →R 02j0, for 1 ≤ j ≤ j′ ≤ r−1

49


02j′1 →R 02j1, for 1 ≤ j ≤ j′ ≤ r−1

02j′1 →L 02j

′+11, for 1 ≤ j′ ≤ r−102j

′1 →L 02j0, for 1 ≤ j ≤ j′ ≤ r−1

02j′0 →L,R 02j0, for 1 ≤ j ≤ j′ ≤ r−2

where L (left) and R (right) identify the topmost ver-tex selected. For rules with annotation L,R, bothvertices can be selected, one at a time. Figs. 68 il-lustrate the idea underlying these rules.

Figure 6: Rewriting 12r1 using the rewrite rules12r1 →L 12r+11 and 12r1 →R 12r+11.

Figure 7: Rewriting 12j′0 using (a) 12j

′0 →R

12j′+10 and (b) a rule 12j

′0 →R 02j0, for 1 ≤ j ≤ j′.

Figure 8: Rewriting 02j′0 using (a) 02j

′0 →L 02j0

and (b) 02j′0 →R 02j0, for some 1 ≤ j ≤ j′.

The correctness and completeness of this rewritesystem can be checked easily by case-analysis, takinginto account the conditions on convexity.

Proposition 1 Let A(r)α,j,β be the number of convex

permutominoes of the class α 2j β with r reex ver-tices, for α, β ∈ 0, 1, 1 ≤ j ≤ r + 1 and r ≥ 0.Then, A

(r)0,j,1 = A

(r)1,j,0, for all r and j (symmetry by

reection w.r.t. V -axis) and A(r)α,j,β is inductively de-

ned as follows.

A(0)1,1,1 = 1

A(r)1,r+1,1 = 2A

(r−1)1,r,1 , for r ≥ 1

A(r)1,j,0 = A

(r−1)1,r,1 +

r−1Xj′=max(1,j−1)

A(r−1)1,j′,0 , for 1 ≤ j ≤ r

A(r)0,j,0 = 2

r−1Xj′=j

A(r−1)1,j′,0 + 2

r−2Xj′=j

A(r−1)0,j′,0 , for 1 ≤ j ≤ r−1

The number of convex permutominoes of size n isgiven by sequence A126020, in [17]. A closed for-mula for this number is given in [8]. In a similar way,we may deduce the recurrences for row-convex permu-tominoes and the simpler classes of bargraphs, stacks,and Ferrers diagrams.

References

[1] T. Auer and M. Held, Heuristics for the generation ofrandom polygons, in: Proc. CCCG'96, 1996, 3843.

[2] E. Barcucci, A. Del Lungo, E. Pergola, and R. Pin-zani, ECO: a methodology for the enumeration ofcombinatorial objects, J. Dier. Equ. Appl. 5 (1999),435490.

[3] G. Barequet and M. Moe, On the complexity ofJensen's algorithm for counting xed polyominoes,J. Discrete Algorithms 5 (2007), 348355.

[4] P. Boldi, V. Lonati, R. Radicioni, and M. Santini,The number of convex permutominoes, Information

and Computation 206 (2008), 10741083.

[5] M. Bousquet-Mélou, Bijection of convex polyominoesand equations for enumerating them according toarea, Discrete Appl. Math. 48 (1994), 2143.

[6] M. Damian, R. Flatland, J. O'Rourke, and S. Ra-maswami, Connecting polygonizations via stretchesand twangs, Theory of Computing Systems 47 (2010),674695.

[7] E. Deutsch, Enumerating symmetric directed convexpolyominoes, Discrete Math. 280 (2004), 225231.

[8] F. Disanto, A. Frosini, R. Pinzani, and S. Rinaldi, Aclosed formula for the number of convex permutomi-noes, Electron. J. Combin. 14 (2007), #R57.

[9] S. Golomb, Polyominoes, Princeton U. Press, 1994.

[10] E. Grazzini, E. Pergola, and M.Poneti, On the ex-haustive generation of convex permutominoes, PureMathematics and Applications 19 (2008), 93104.

[11] D. Hickerson, Counting horizontally convex polyomi-noes, J. Integer Sequences 2 (1999), Article 99.1.8.

[12] F. Insitti, Permutation diagrams, xed points andKazhdan-Lusztig R-polynomials, Annals of Combi-

natorics 10 (2006), 369387.

[13] I. Jensen, Enumerations of lattice animals and trees,J. Stat. Phys. 102 (2001), 865881.

[14] I. Jensen, Counting polyominoes: a parallel imple-mentation for cluster computing, in: Proc. ICCS'03,LNCS, vol. 2659, Springer, 2003, 203212.

[15] A. Del Lungo, E. Duchi, A. Frosini, and S. Rinaldi,On the generation and enumeration of some classes ofconvex polyominoes, Electron. J. Combin. 11 (2004),#R60.

[16] J. O'Rourke, An alternate proof of the rectilinear artgallery theorem, J. Geometry 21 (1983), 118130.

[17] N. J. A. Sloane, The On-Line encyclopedia of integersequences, OEIS Foundation, http://oeis.org/.

[18] C. Sohler, Generating random star-shaped polygons,in: Proc. CCCG'99, 1999.

[19] A. P. Tomás and A. Bajuelos, Quadratic-time linear-space algorithms for generating orthogonal polygonswith a given number of vertices, in: Proc. CGA'04

ICCSA'04, LNCS, vol. 3045, Springer, 2004, 117126.

[20] C. Zhu, G. Sundaram, J. Snoeyink, and J. S. B. Mit-chell, Generating random polygons with given ver-tices, Comput. Geom. 6 (1996), 277290.

50


Distance domination, guarding and vertex cover for maximal

outerplanar graphs

Santiago Canales1, Gregorio Hernández∗2, Mafalda Martins‡†3, and Inês Matos‡3

1Universidad Ponticia Comillas de Madrid, Spain2Universidad Politécnica de Madrid, Spain3Universidade Aveiro & CIDMA, Portugal

Abstract

In this paper we dene a distance guarding concept onplane graphs and associate this concept with distancedomination and distance vertex cover concepts on tri-angulation graphs. Furthermore, for any n-vertexmaximal outerplanar graph, we provide tight up-per bounds for g2d(n) (2d-guarding number), γ2d(n)(2d-domination number) and β2d(n) (2d-vertex covernumber).

Introduction

Domination, covering and guarding are widely studiedsubjects in graph theory. Given a graph G = (V,E)a dominating set is a subset D of V such that ev-ery vertex not in D is adjacent to a vertex in D. Asubset C of V is a vertex cover if each edge of thegraph is incident to at least one vertex of the set.Thus, a dominating set guards the vertices of a graphwhile a vertex cover guards its edges. In plane graphsthese concepts dier from the guarding set conceptthat guards the faces of the graph. Let G = (V,E) bea plane graph, a guarding set is a subset S of V suchthat every face has a vertex in S. There are many pa-pers and books about domination and its many vari-ants in graphs, e.g. [4, 8, 9, 10]. Domination was ex-tended to distance domination by Meir and Moon [11]:given a graph G, a subset D of V is said to be a dis-tance k-dominating set if for each vertex u ∈ V −D,distG(u, v) ≤ k for some v ∈ D. The minimum car-dinality of a distance k-dominating set is said to bethe distance k-domination number of G and is de-noted by γkd(G). In the case of distance domination,

∗Research supported by ESF EUROCORES programme Eu-roGIGA - ComPoSe IP04 - MICINN Project EUI-EURC-2011-4306.

†Email: [email protected]. Research also supportedby FCT grant SFRH/BPD/66431/2009.

‡Research supported by FEDER funds throughCOMPETEOperational Programme Factors of Com-petitiveness, CIDMA and FCT within project PEst-C/MAT/UI4106/2011 with COMPETE number FCOMP-01-0124-FEDER-022690.

there are also some known results concerning boundsfor γkd(G), e.g., [13, 14]. However, if graphs are re-stricted to triangulations, as far as we know, thereare no known bounds for γkd(G). The distance domi-nation was generalized to broadcast domination whenthe power of each vertex may vary [6]. Given a graphG = (V,E), a broadcast is a function f : V → N0. Abroadcast is dominating if for every vertex v, thereis a vertex u such that f(u) > 0 and d(u, v) ≤ f(u).A dominating broadcast f is optimal if

∑v∈V f(v)

is minimum over all choices of broadcast dominatingfunctions for G. The broadcast domination problemconsists in building this optimal function. Note that,if f(V ) = 0, k, then the broadcast domination prob-lem is the distance k-dominating problem. If a broad-cast f provides coverage to the edges of G instead ofcovering its vertices, we have a generalization of thevertex cover concept [2]. A broadcast f is covering iffor every edge (x, y) ∈ E, there is a path P in G thatincludes the edge (x, y) and one end of P must be avertex u, where f(u) is at least the length of P . A cov-ering broadcast f is optimal if

∑v∈V f(v) is minimum

over all choices of broadcast covering functions for G.Note that, if f(V ) = 0, 1, then the broadcast coverproblem coincides with the problem of nding a min-imum vertex cover. Regarding the broadcast coverproblem when all vertices have the same power (i.e.,when f(V ) = 0, k, for a xed k = 1), as far as weknow, there are no published results besides [5] wherethe authors propose a centralized and distributed ap-proximation algorithm to solve it. Concerning theguarding concept there are also known bounds on theguarding number g(G). For example, g(G) ≤ ⌊n

2 ⌋, forany n-vertex plane graph [3], and if G is a maximalouterplanar graph this bound is ⌊n

3 ⌋ [7]. Contrary tothe notions of domination and vertex cover on planegraphs that were extended to include its distance ver-sions, the guarding concept was not generalized to itsdistance version.

In this paper we generalize the guarding concepton plane graphs to its distance guarding version. Wealso formalize the broadcast cover problem when allvertices have the same power, which we call distance

51


k-vertex cover. Since there are no combinatorial re-sults for the concepts of domination, vertex coverand guarding on its distance versions on triangulationgraphs (triangulations, for short), we also address thisproblem. In the next section we describe some of theterminology that will be used throughout this paper.

1 Preliminaries

Given a triangulation of a point set T = (V,E), wesay that a bounded face Ti of T (i.e., a triangle) is kd-visible from a vertex p ∈ V , if there is a vertex x ∈ Ti

such that distT (x, p) ≤ k − 1. The kd-visibility regionof a vertex p ∈ V comprises the triangles of T that arekd-visible from p (see Fig. 1).

p p

( )a (b)

Figure 1: The kd-visible region of p for: (a) k = 1;(b) k = 2.

A kd-guarding set for T is a subset F ⊆ V such thatevery triangle of T is kd-visible from an element of F .We designate the elements of F by kd-guards. The kd-guarding number gkd(T ) is the number of vertices ina smallest kd-guarding set for T . Note that, to avoidconfusion with multiple guarding [1] - where the typ-ical notation is k-guarding - we will use kd-guarding,with an extra d. Given a set S and a positiveinteger n, we dene gkd(S) = maxgkd(T ) : T =(V,E) is triangulation with V = S and gkd(n) =maxgkd(S) : |S| = n. A kd-vertex cover for T , ordistance k-vertex cover for T , is a subset C ⊆ V suchthat for each edge e ∈ E there is a path of length atmost k, which contains e and a vertex of C. Thekd-vertex cover number βkd(T ) is the number of ver-tices in a smallest kd-vertex cover set for T . Givena set S and a positive integer n, we dene βkd(S) =maxβkd(S) : T = (V,E) is triangulation with V =S and βkd(n) = maxβkd(S) : |S| = n. Finally,as already dened by other authors, a kd-dominatingset for T , or distance k-dominating set for T , isa subset D ⊆ V such that each vertex u ∈ V −D,distT (u, v) ≤ k for some v ∈ D. The kd-dominationnumber γkd(T ) is the number of vertices in a smallestkd-dominating set for T . Given a set S and a positiveinteger n, we dene γkd(S) = maxγkd(T ) : T =(V,E) is triangulation with V = S and γkd(n) =maxγkd(S) : |S| = n. Our main goal is to obtainbounds on γkd(T ), gkd(T ) and βkd(T ). We start by

establishing a tight upper bound for g2d(n) for a spe-cial class of triangulations, namely the maximal out-erplanar graphs. A graph is a maximal outerplanargraph if it is a triangulation of a simple polygon with-out holes [12]. Edges on the exterior face are calledexterior edges, and interior edges otherwise.In the next section we show that there is a re-

lationship between 2d-guarding, 2d-dominating and2d-vertex cover sets on triangulations. In sections 3and 4 we provide upper bounds for g2d(n), γ2d(n) andβ2d(n) on maximal outerplanar graphs and show thatthese bounds are tight.

2 Relationship between distance

vertex cover, guarding and

domination on triangulations

We start by showing that the three concepts are dif-ferent. Fig. 2 depicts 2d-dominating and 2d-guardingsets for a given triangulation. Note that in Fig. 2(a)the set u, v is 2d-dominating since the remainingvertices are at distance 1 or 2. However, it is nota 2d-guarding set because the shaded triangle is notguarded, as its vertices are at distance 2 from u, v.In 2(b) w, z is a 2d-guarding set, however it is not a2d-vertex cover since any path between the bold edgeand w or z has length at least 3. Therefore, the boldedge is not covered.

uv

w

z

( )a (b)

Figure 2: (a) 2d-dominating set for a triangulation T ;(b) 2d-guarding set for T .

Now we are going to establish a relation betweeng2d(T ), γ2d(T ) and β2d(T ). The following results canbe easily generalized to gkd(T ), γkd(T ) and βkd(T ).

Lemma 1 If C is a 2d-vertex cover of a triangulationT , then C is a 2d-guarding set and a 2d-dominatingset of T .

Lemma 2 If F is a 2d-guarding set of a triangulationT , then F is a 2d-dominating set of T .

The previous lemmas prove the following result.

Theorem 3 Given a triangulation T the minimumcardinality g2d(T ) of any 2d-guarding set veriesγ2d(T ) ≤ g2d(T ) ≤ β2d(T ).

52


(a) (b) (c)

Figure 3: (a) A 2d-dominating set for a triangulation T (black vertices); (b) a 2d-guarding set for T (grayvertices); (c) each of the bold edges needs a dierent vertex to be 2d-covered.

Note that the inequalities above can be strict, aswe can see in the triangulation T presented in Fig. 3,where γ2d(T ) = 2, g2d(T ) = 3, and β2d(T ) ≥ 4.

3 2d-domination and 2d-guarding of maximal out-

erplanar graphs

In this section we establish upper bounds for g2d(n)and γ2d(n) on maximal outerplanar graphs. In orderto do this, and following the ideas of O'Rourke [12],we rst need to introduce some lemmas.

Lemma 4 Suppose that f(m) guards are always suf-cient to 2d-guard any outerplanar maximal graph Gwith m vertices. Then if G has two guards placedat any two adjacent vertices or one guard placed atany one of its vertices, then f(m− 2) or f(m− 1)additional guards are sucient to 2d-guard G, respec-tively.

Lemma 5 Let G be an outerplanar maximal graphwith n ≥ 2k vertices. There is an interior edge eof G that partitions G into two pieces, one of whichcontains m = k, k + 1, . . . , 2k − 1 or 2k − 2 exterioredges of G.

Theorem 6 Every n-vertex maximal outerplanargraph, with n ≥ 5, can be 2d-guarded by ⌊n

5 ⌋ 2d-guards. That is, g2d(n) ≤ ⌊n

5 ⌋ for all n ≥ 5.

Proof. For 5 ≤ n ≤ 11, the truth of the theoremcan be easily established. It should be noted thatfor n = 5 the 2d-guard can be placed randomly andfor n = 6 it can be placed at any vertex of degreegreater than 2. Assume that n ≥ 12 and that thetheorem holds for all n′ < n. Lemma 5 guaranteesthe existence of an interior edge e that divides G intotwo maximal outerplanar graphsG1 andG2, such thatG1 has m exterior edges of G with 6 ≤ m ≤ 10. Thevertices of G are labeled with 0, 1, . . . , n− 1 such thate is (0,m). Each value of m is considered separately.We present the cases m = 6 and m = 9.

Case m = 6. G1 has m+ 1 = 7 exterior edges, thusG1 can be 2d-guarded with one guard. G2 has n − 5

exterior edges including e, and by induction hypothe-sis, it can be 2d-guarded with ⌊n−5

5 ⌋ = ⌊n5 ⌋−1 guards.

Thus G1 and G2 together can be 2d-guarded by ⌊n5 ⌋

guards.

Case m = 9. The presence of any of the internal edges(0,8), (0,7), (0,6), (9,1), (9,2) and (9,3) would violatethe minimality of m. Thus, the triangle T in G1 thatis bounded by e is either (0,5,9) or (0,9,4). Since theseare equivalent cases, suppose that T is (0,5,9) (see Fig.4(a)).

0 9

7

6

54

3

2

1 8

e

G2

T

G1

0 9

7

6

54

8

e

G2

3

2

1T

G1

( )a ( )b

Figure 4: (a) The triangle T is (0,5,9); (b) the internaledge (0,4) and the triangle (6,7,8) are present.

The pentagon (5,6,7,8,9) can be 2d-guarded by plac-ing one guard at a chosen vertex. However, to 2d-guard the hexagon (0,1,2,3,4,5) we cannot place aguard randomly.We will consider two separate cases:(i) The internal edge (0,4) is not present: if a guardis placed at vertex 5, then the hexagon (0,1,2,3,4,5)is 2d-guarded, thus G1 is 2d-guarded. Since G2 hasn− 8 edges it can be 2d-guarded by ⌊n−8

5 ⌋ ≤ ⌊n5 ⌋ − 1

guards applying the induction hypothesis. This yieldsa 2d-guarding of G by ⌊n

5 ⌋ guards; (ii) The internaledge (0,4) is present: if a guard is placed at vertex0, then G1 is 2d-guarded unless the triangle (6,7,8)is present in the triangulation (see Fig. 4(b)). Inany case, two 2d-guards placed at vertices 0 and 9guard G1. G2 has n − 8 exterior edges, includinge. By lemma 4 the two guards placed at vertices 0and 9 allow the remainder of G2 to be guarded byf(n− 8− 2) = f(n− 10) additional 2d-guards. Recallthat f(n′) is the number of 2d-guards that are alwayssucient to guard a maximal outerplanar graph with

53


n′ vertices. By the induction hypothesis f(n′) = ⌊n′

5 ⌋.Thus, ⌊n−10

5 ⌋ = ⌊n5 ⌋ − 2 guards suces to 2d-guard

G2. Together with the guards placed at vertices 0and 9 that 2d-guard G1, all G is guarded by ⌊n

5 ⌋ 2d-guards.

To prove that this upper bound is tight we needto construct a maximal outerplanar graph G of or-der n such that g2d(G) = ⌊n

5 ⌋. Fig. 5 shows a maxi-mal outerplanar graph G for which γ2d(G) = n

5 , sincethe black vertices can only be 2d-dominated by dif-ferent vertices. Thus, γ2d(n) ≥ n

5 . Note that thisexample can be generalized to kd-domination to ob-tain γkd(n) ≥ n

(2k+1) .

Figure 5: A maximal outerplanar graph G for whichγ2d(G) = n

5 .

According to theorem 3, γ2d(G) ≤ g2d(G), so⌊n5 ⌋ ≤ g2d(G). In conclusion, ⌊n

5 ⌋ 2d-guards are oc-casionally necessary and always sucient to guard an-vertex maximal outerplanar graph G. On the otherhand, we can also establish that γ2d = ⌊n

5 ⌋, since⌊n5 ⌋ ≤ γ2d(n) and γ2d(n) ≤ g2d(n), for all n. Thus,

it follows:

Theorem 7 Every n-vertex maximal outerplanargraph with n ≥ 5 can be 2d-guarded (and 2d-dominated) by ⌊n

5 ⌋ guards. This bound is tight.

4 2d-covering of maximal outer-

planar graphs

In this section we determine an upper bound for 2d-vertex cover on maximal outerplanar graphs which isalso tight.

Lemma 8 Suppose that f(m) vertices are always suf-cient to 2d-cover any outerplanar maximal graph Gwith m vertices. If we randomly choose a vertex of Gto be a 2d-covering vertex, then f(m − 1) additionalvertices are sucient 2d-cover G.

Theorem 9 Every n-vertex maximal outerplanargraph, with n ≥ 4, can be 2d-covered with ⌊n

4 ⌋ ver-tices. That is, β2d(n) ≤ ⌊n

4 ⌋ for all n ≥ 4.

Now, we will prove that this upper bound is tight.The bold edges of the maximal outerplanar graph il-lustrated in Fig. 6 can only be 2d-covered from dier-ent vertices, and therefore β2d(n) ≥ n

4 . To conclude:

Theorem 10 Every n-vertex maximal outerplanargraph with n ≥ 5 can be 2d-covered by ⌊n

4 ⌋ vertices.This bound is tight.

Figure 6: A maximal outerplanar graph G for whichβ2d(G) = n

4 .

References

[1] P. Belleville, P. Bose, J. Czyzowicz, J. Urrutia, J.Zaks, K-vertex guarding simple polygons, Comput.

Geom. Theory Appl. , 42(4) (2009), 352361.

[2] J.R.S. Blair and S.B. Horton, Broadcast covers ingraphs, Congr. Num. 173 (2005), 109115.

[3] P. Bose, T. Shermer, G. Toussaint, and B. Zhu,Guarding Polyhedral Terrains, Computational Geom-

etry: Theory and Applications, 6(3)(1997), 173185.

[4] C.N. Campos and Y. Wakabayashi, On dominat-ing sets of maximal outerplanar graphs, Discrete

Appl.Math, 161(3) (2013), 330335.

[5] Q. Chen, L. Zhao, Approximation algorithms forthe L-distance vertex cover problem, Proc. ICTMF

(2012), Bali, Indonesia, 100104.

[6] D. Erwin, Dominating broadcasts in graphs, Bull.

Inst. Combin. Appl. 42 (2004), 89105.

[7] S. Fisk. A short proof of Chvatal's watchman theo-rem, J. Combin. Theory Ser. B, (24) (1978), 374.

[8] T.W. Haynes, S.T. Hedetniemi, P.J. Slater, Funda-mentals of Domination in Graphs, M. Dekker, 1998.

[9] E. King, J. Pelsmajer, Dominating sets in plane tri-angulations, Discret Math. 310(17-18) (2010), 22212230.

[10] L.R. Matheson, R.E. Tarjan, Dominating sets in pla-nar graphs, Eur. J. Combin. 17(6) (1996), 565568.

[11] A. Meir, J.W. Moon, Relations between packing andcovering number of a tree, Pacic J. Math 61(1)(1975), 225233.

[12] J. O'Rourke, Art Gallery Theorems and Algorithms,Oxford University Press, Inc., New York, USA, 1987.

[13] F. Tian, J. Xu, Bounds for distance domination num-bers of graphs, Journal of University of Science and

Technology of China 34(5) (2004), 529534.

[14] F. Tian, J. Xu, A note on distance domination num-bers of graphs, Aust. J. of Combin 43 (2009), 181190.

54



Rolf Klein∗

Universität Bonn, Institut für Informatik I

Abstract

Abstract Voronoi diagrams are a unifying frameworkthat covers many types of concrete Voronoi diagrams.This talk reports on the state of the art, includingrecent progress.

Introduction

Concrete Voronoi diagrams [1] are mostly dened interms of sites and distance, and both concepts canvary greatly. Abstract Voronoi diagrams [6] are builton what most concrete diagrams have in common: asystem of simple bisecting curves J(p, q) = J(q, p),where p, q are just indices from a set S of n elements.Each curve J(p, q) divides the plane into two domains,D(p, q) and D(q, p). The abstract Voronoi region of pwith respect to S is dened by

VR(p, S) :=⋂

q∈S\p

D(p, q)

and the abstract Voronoi diagram of S is just the planeminus all Voronoi regions.An interesting question is what properties to re-

quire of the curves J(p, q). They should be as weakas possible for generality, but strong enough to en-sure that useful Voronoi structures result from theabove denitions. It turns out [7] that the followingare sucient.

(A1) Each curve J(p, q), where p 6= q, is unbounded.After stereographic projection to the sphere,it can be completed to a closed Jordan curvethrough the north pole.

For any three indices p, q, r in S, and S′ :=p, q, r,

(A2) each Voronoi region VR(p, S′) is path-wise con-nected,

(A3) each point of the plane belongs to the closure ofa Voronoi region VR(p, S′).

∗Email: [email protected]. This work was supportedby the European Science Foundation (ESF) in the EURO-CORES collaborative research project EuroGIGA/VORONOI.

p r

pq

qr

p rpq q r

pq

pr

rq

prp qq r

Figure 1: Admissible curve systems

Informally, if the bisecting curves are un-bounded and behave decently, and if any tripletJ(p, q), J(p, r), J(q, r) is situated as shown in Fig-ure 1, the AVD theory applies.

1 Results

This means that structural results and ecient algo-rithms become available without further eort [7].

Theorem 1 V (S) is a planar graph of complexity

O(n). It can be constructed in an expected number

of O(n log n) many steps.

If we replace Axiom A2 by the more general re-quirement

(A2') Each Voronoi region VR(p, S′) has at most s con-nected components

(and assume that any two curves intersect onlynitely often), the above result can be generalized asfollows [3].

Theorem 2 Abstract Voronoi diagrams with discon-

nected regions can be computed in an expected number

of

O

s2nn∑

j=3

mj

j

steps, where mj denotes the average number of faces

per region in all AVDs of j sites from S.

55


One can extend the denition of abstract Voronoidiagrams to orders k > 1 by dening

VRk(P, S) :=⋂

p∈P, q∈S\P

D(p, q).

For order k = n−1, the resulting AVDs are trees [9] oflinear size. In the general case the following complex-ity result holds. Here we assume that all curves arein general position, and that the standard Voronoi-regions are non-empty.

Theorem 3 The abstract order-k Voronoi diagram

V k(S) has at most 2k(n− k) many faces. This bound

can be achieved.

2 Conclusion

Open is the case of closed bisecting curves.

References

[1] F. Aurenhammer, R. Klein, and D.-T. Lee, VoronoiDiagrams and Delaunay Triangulations, World Sci-entic Publishing Company, to appear August 2013.

[2] C. Bohler, P. Cheilaris, R. Klein, C.H. Liu, E. Pa-padopoulou, and M. Zavershynskyi, On the complex-ity of higher order abstract Voronoi diagrams, to bepresented at ICALP' 13.

[3] C. Bohler and R. Klein, Abstract Voronoi diagramswith disconnected regions, Bonn 2013, manuscript.

[4] C. Bohler and R. Klein, Point sites with individualdistance functions, Bonn 2012, manuscript.

[5] A. G. Corbalan, M. Mazon, and T. Recio, Geometryof bisectors for strictly convex distance functions, In-ternational Journal of Computational Geometry and

Applications 6(1) (1996), 4558.

[6] R. Klein, Concrete and abstract Voronoi diagrams,Lecture Notes in Computer Science, 400, Springer-Verlag, 1987.

[7] R. Klein, E. Langetepe, and Z. Nilforoushan, Ab-stract Voronoi diagrams revisited, Computational

Geometry: Theory and Applications 42(9) (2009),885902.

[8] R. Klein, K. Mehlhorn, and S. Meiser, Random-ized incremental construction of abstract Voronoi di-agrams, Computational Geometry: Theory and Ap-

plications 3 (1993), 157184.

[9] K. Mehlhorn, S. Meiser, and R. Rasch, Furthestsite abstract Voronoi diagrams, International Journalof Computational Geometry and Applications 11(6)(2001), 583616.

56



Pedro Ramos∗1 and William Steiger†2

1Department of Physics and Mathematics, University of Alcalá, Alcalá de Henares, Spain.2Department of Computer Science, Rutgers University, 110 Frelinghuysen Road, Piscataway, New Jersey

08854-8004.

Abstract

An intriguing conjecture of Nandakumar and RamanaRao is that for every convex body K ⊆ R2, and forany positive integer n, K can be expressed as theunion of n convex sets with disjoint interiors and eachhaving the same area and perimeter. The rst dicultcase - n = 3 - was settled by Bárány, Blagojevi¢, andSzucs using powerful tools from algebra and equivari-ant topology. Here we give an elementary proof ofthis result in case K is a triangle, and show how toextend the approach to prove that the conjecture istrue for triangles.

Introduction

Let K be a convex body in the plane. Nandakumarand Ramana Rao [7] noticed that if a ham-sandwichcut for K were rotated through π radians - alwaysmaintaining a bisection of K - then at some point inthis process, K is partitioned into two convex partswith disjoint interiors, and each having the same areaand perimeter. A slightly more careful argument us-ing this fact, along with induction, was given to showthat for n = 2k, K can always be partitioned into nconvex subsets, each with the same area and perime-ter. They made the intriguing

Conjecture 1 For every n ∈ N and all convex bodiesK ⊆ R2, K is the disjoint union of n convex pieces,each with the same area and perimeter.

The conjecture describes an n−equipartition ofK (be-cause of the n equal areas) which is in addition fair,by virtue of the equal perimeters. Observe that, inthe related problems of cake partitioning [1] only theperimeter of ∂K is taken into account.

∗Email: [email protected]. Partially supported by MEC

grant MTM2011-22792 and by the ESF EUROCORES pro-

gramme EuroGIGA, CRP ComPoSe, under grant EUI-EURC-

2011-4306.†Email: [email protected]. Work supported in part by

grant 0944081, National Science Foundation, USA. The author

thanks The University of Alcala for supporting a visit to their

Department of Mathematics. He is grateful to that department

for their congenial hospitality.

Bárány et. al. [3], using heavy-duty tools from alge-bra and equivariant topology settled the case n = 3:A 3−fan is a point P in the plane with three raysemanating from it. It is convex if all angles are atmost π. It equipartitions K if the three rays divideK into three regions of equal area, and it is fair, ifthese regions also have equal perimeter. Bárány et.al. showed that there is a convex 3−fan that makesa fair equi-partition of K. Subsequently, Aronov andHubard [2] and then Karasev [6], showed that the con-jecture was true for n = pk, a prime power, and alsoin dimension d ≥ 2, with area replaced by volumeand perimeter, by surface area. Blagojevi¢ andZiegler found some problems with the proofs in thesetwo papers, so they established the results - and more- using dierent tools. In the present paper, in an at-tempt to understand some of the geometric featuresof this problem and why - or why not - it may bedicult, we use (only) elementary methods to studythe conjecture for R2, and when K is a triangle. Wecall a 3−fan interior for K if the apex P is interior toK; otherwise it is exterior. In the rst case, all threerays play a role in the partition. In the second case,the partition is simply via two chords (which mightmeet on the boundary of K, but not in its interior).Because our results concern only exterior 3-fans, wewill state them using chords. A main result of thispaper is

Theorem 1 Every triangle has a fair equi-partitiondened by two disjoint chords.

More importantly, the ideas used to prove this re-sult can be extended to show that every triangle hasa fair n-partition dened by n− 1 disjoint chords.

Corollary 2 For every n ≥ 1, every triangle has afair, n-equipartition using n− 1 disjoint chords.

In [7] it was observed that when K is a triangle,it can be covered by n = k2 disjoint triangles simi-lar to K, and all with the same area (and perimeter).But Theorem 1 is a new step toward resolving Con-jecture 1, if only for triangles.

57


A P P ′

Q

Q′

B

C

Figure 1: Illustration for Lemma 2.

1 Some details

In this section we describe some ideas behind theproofs for our results. The full proofs will appear inthe actual paper.We are able to understand this problem in the tri-

angle case partly due to a simple tool that describeshow perimeters change when a chord in a partition ismoved slightly while still preserving the areas of allregions.We use a Cartesian coordinate system in the Eu-

clidean plane, and denote by AB and |AB|, respec-tively, the segment dened by points A and B and its

length. Vector−−→OP and point P will be identied if the

context is clear enough. The list of points AB · · ·Dwill be used to denote the corresponding polygon and,nally, π(·) will be the perimeter of the polygon.Let A, B, C be non collinear points, and x A as

the origin of the coordinate system.

Lemma 3 Consider points P ∈ AB and Q ∈ ACsuch that |AP | ≤ |AQ|. Let P ′ = tP and let Q′ = 1

tQ(then the area of APQ equals the area of AP ′Q′).

1. π(AP ′Q′) and π(BCQ′P ′) are convex functionsof t, achieving their minima when |AP ′| = |AQ′|.

2. If we write ∆π1 = π(AP ′Q′) − π(APQ) and∆π2 = π(BCQ′P ′) − π(BCQ′P ′) then |∆π1| ≥|∆π2|.

Consider a unit area triangle ∆ = ABC. Withoutloss of generality, we can assume that the smallestside is AB and that the coordinates of its vertices areA = (0, 0), B = (b, 0), and C = (c, 2/b), with b > 0and c ≥ b/2, as in the gure above. If we take pointsU = (b/3, 0) and V = (2b/3, 0), ∆ is partitioned into∆1 = CAU,∆2 = CUV , and ∆3 = CV B, all with thesame area. The goal is to rotate the chords CU andCV maintaining equality of all three areas, but in sucha way as to force all three perimeters to coincide. Todescribe this process unambiguously, we place pointsD and E on the boundary of ∆. Initially both points

U B

C

A

D

V

E

∆1 ∆3

∆2

Figure 2: Rotation for an isosceles triangle.

are placed on C. We then manipulate the chords DUand EV by moving the endpoints along the boundaryof ∆, thus altering the three sets in the partition. Weuse the notation πi to denote the perimeter of regioni.

The case where x = b/2, and ∆ is isosceles, iseasiest. Here, we move chord DU counter-clockwise(maintaining the area of ∆1 = DAU), and chord EVclockwise (maintaining the area of ∆3 = EV B) un-til U and V coincide at F = (b/2, 0) (see Figure 2).During this process the middle region is a pentagon∆2 = CDUV E, ending at ∆2 = CDFE. If we alsokeep U+V = (b, 0), there will be a position where thepartition is fair, by the intermediate value theorem,since ∆1 and ∆3 initially have equal perimeters, butlarger than that of ∆2, and at the end, π1 = π3 ≤ π2,by virtue of AB being the smallest side of ∆.

When ∆ is not isosceles, we rotate DU , again main-taining the area, till the perimeters of two regions areequal. Observe that, during the rotation, both π1 andπ2 decrease. If b/2 < c < 2b/3, because we start withperimeters π1 > π3 > π2, we must reach case 1 inFigure 3, where π1 = π3 > π2. If c ≥ 2b/3, we startwith perimeters π1 > π2 ≥ π3. When rotating DU wecan get π2 = π3 < π1 (case 2) or π1 = π2 > π3 (case3).

Using Lemma 3 we know how to proceed in eachcase. For case 1, we rotate the chord EV clockwise,and the chordDU counterclockwise, preserving equal-ity for the areas of regions 1 and 3. For case 2, werotate the chords in the same way, maintaining nowequal areas for regions 2 and 3. Finally, for case 3both chords are rotated counterclockwise, keeping ar-eas of regions 1 and 2 equal. Theorem 1 will be provenif we show that during this rotation, equality of thethree perimeters is achieved.

For cases 1 and 2, the key is to observe that the footof the left chord reaches the midpoint of AB rst. If Uis the midpoint of AB, we consider the triangle UBE′,congruent with AUD (see Figure 4,left). From this,it is not hard to see that ∆3 has to be V BE, withV ∈ UB and that π1 < π2.

For case 3, consider a partition into three pieces of

58


Case 2: π2 = π3 < π1

∆1 ∆2 ∆3

Case 1: π1 = π3 > π2

Case 3: π1 = π2 > π3

A B A B

C C

V V

∆1 ∆2 ∆3

A B

C

V

∆1 ∆2 ∆3

Figure 3: After the rst rotation, two perimeters areequal.

U B

C

A

D E′

V

E

∆1 ∆3

∆2

∆3

E

V B

C

A U

D

∆2

∆1

Figure 4: Situation at the end of the rotation.

equal area (but dierent perimeters) using two chordsparallel to BC (see Figure 4,right). In this setting, itcan be shown that π1 + π2 < 2π3. Therefore, if weconsider now the chord that divides AV E into twopieces of equal area and perimeter, from Lemma 3 itfollows that perimeters of regions 1 and 2 are smallerand in that situation we must have π1 = π2 < π3.

The approach of the arguments above can be ap-plied to prove the existence of a fair n-equipartition ofevery triangle. Take points Vi = (ib/n, 0) and pointsUi, i = 0, . . . , n; initially all Ui = C. The n−1 chordsUiVi, i = 1, . . . , n − 1 partition ∆ into n triangles(∆i = Ui−1Vi−1Vi, i = 1, . . . , n), of equal area. Weorder the chords left to right, and so the chord withfoot at Vi is the i-th chord. The chords for whichVi < c are called left chords, and the rest are the rightchords.

We start by rotating the rst chord counterclock-wise, until the perimeter of the rst region is equal,either to the perimeter of the second region, or tothe perimeter of the last one. In the rst case, we

continue by rotating the rst and the second chord(both counterclockwise), maintaining equality of theperimeters of the rst two regions. In the second case,we continue by rotating the rst chord counterclock-wise, and the last chord clockwise, again maintainingequal perimeters for the involved regions. This pro-cess can be iterated, and when we reach the centralregion (the one in between the last left chord and therst right chord), we will have cases analogous to theones in Figure 3. We proceed in the same way here,and it can be shown than equality of all perimetersis obtained before reaching a critical situation. Theproof of this fact is more involved and will appear inthe full version of this paper.

2 Discussion

It is clear that an equilateral triangle has two distinctfair 3-equipartitions that are interior (as well as threeexterior ones - for each vertex, the process outlined inthe previous section produces an equi-partition withan exterior 3-fan). It is easy to see that skinny tri-angles do not have fair interior 3-equipartitions. Itwould be interesting to characterize which triangleshave equipartitions both exterior and interior, andwhich have only exterior ones.It is easy to see that fair equipartitions of disks have

to be radial, with the vertex at the center. Therefore,we cannot expect to extend the approach of this paperto the general case. Nevertheless, some convex bod-ies may have fair n-equipartitions produced by n− 1disjoint chords. It would be interesting to try to char-acterize this family.

References

[1] J. Akiyama and A. Kaneko and M. Kano and G. Naka-mura and E. Rivera-Campo and S. Tokunaga andJ. Urrutia. Radial perfect partitions of convex setsin the plane". Discrete and computational geometry(Tokyo, 1998), 1-13, Lecture Notes in Comput. Sci.,1763, Springer, Berlin, 2000.

[2] B. Aronov and A. Hubard. Convex equipartitions ofvolume and surface area. arXiv.org/abs/1010.4611,2010.

[3] I. Bárány, I., P. Blagojevi¢, and A. Szucs. Equiparti-tioning by a convex 3-fan. Advances in Mathematics,223 (2), 579-593, 2010.

[4] I. Bárány, I., P. Blagojevi¢, and A. Blagojevi¢ Func-tions, measures, and equipartitioning k-fans. Discreteand Computational Geometry, 2013, to appear.

[5] P. Blagojevi¢ and G. Ziegler. Convex equipar-titions by equivariant obstruction theory.arXiv:1202.5504v2, 2012.

[6] R. Karasev. Equipartition of several measures.arXiv:1011.4762, 2010.

59


[7] R. Nandakumar and N. Ramana Rao 'Fair' parti-tions of polygons' - an introduction Proc. IndianAcademy of Sciences - Mathematical Sciences (to ap-pear); http://arxiv.org/abs/0812, 2010.

60

XV Spanish Meeting on Computational Geometry, June 26-28, 2013On the nonexisten e of k-reptile simpli es in R3 and R4 ∗Jan Kyn£l†1 and Zuzana Safernová‡11 Department of Applied Mathemati s and Institute for Theoreti al Computer S ien e, Charles University,Fa ulty of Mathemati s and Physi s, Malostranské nám. 25, 118 00 Praha 1, Cze h Republi Abstra tA d-dimensional simplex S is alled a k-reptile (or ak-reptile simplex ) if it an be tiled without overlaps byk simpli es with disjoint interiors that are all mutu-ally ongruent and similar to S. For d = 2, triangulark-reptiles exist for many values of k and they havebeen ompletely hara terized by Snover, Waiveris,and Williams. On the other hand, the only k-reptilesimpli es that are known for d ≥ 3, have k = md,where m is a positive integer. We substantially sim-plify the proof by Matou²ek and the se ond authorthat for d = 3, k-reptile tetrahedra an exist only fork = m3. We also prove a weaker analogue of thisresult for d = 4 by showing that four-dimensional k-reptile simpli es an exist only for k = m2.1 Introdu tionA losed set X ⊂ Rd with nonempty interior is alled a k-reptile (or a k-reptile set) if there aresets X1, X2, . . . , Xk with disjoint interiors and withX = X1 ∪X2 ∪ · · · ∪Xk that are all mutually ongru-ent and similar to X . Su h sets have been studied in onne tion with fra tals and also with rystallographyand tilings of Rd [4, 8, 9, 11.It easy to see that whenever S is a d-dimensionalk-reptile simplex, then all of Rd an be tiled by on-gruent opies of S: indeed, using the tiling of S by itssmaller opies S1, . . . , Sk as a pattern, one an indu -tively tile larger and larger similar opies of S. Onthe other hand, not all spa e-lling simpli es must bek-reptiles for some k ≥ 2.Clearly, every triangle tiles R2. Moreover, everytriangle T is a k-reptile for k = m2, sin e T an betiled in a regular way with m2 ongruent tiles, ea h

∗The authors were supported by the proje t CE-ITI (GACRP202/12/G061) of the Cze h S ien e Foundation, by the grantSVV-2013-267313 (Dis rete Models and Algorithms) and byproje t GAUK 52410. The resear h was partly ondu ted dur-ing the Spe ial Semester on Dis rete and Computational Geom-etry at É ole Polyte hnique Féderale de Lausanne, organizedand supported by the CIB (Centre Interfa ultaire Bernoulli)and the SNSF (Swiss National S ien e Foundation).†Email: kyn lkam.m. uni. z‡Email: zuzkakam.m. uni. z

positively or negatively homotheti to T . See e. g.Snover et al. [19 for an illustration.The question of hara terizing the tetrahedra thattile R3 is still open and apparently rather di ult.The rst systemati study of spa e-lling tetrahedrawas made by Sommerville. Sommerville [20 dis ov-ered a list of exa tly four tilings (up to isometry andres aling), but he assumed that all tiles are prop-erly ongruent (that is, ongruent by an orientation-preserving isometry) and meet fa e-to-fa e. Ed-monds [6 noti ed a gap in Sommerville's proof andby ompleting the analysis, he onrmed that Som-merville's lassi ation of proper, fa e-to-fa e tilingsis omplete. In the non-proper and non fa e-to-fa esituations there are innite families of non-similartetrahedral tilers. Goldberg [10 des ribed three su hfamilies, obtained by partitioning a triangular prism.In fa t, Goldberg's rst family was found by Som-merville [20 before, but he sele ted only spe ial aseswith a ertain symmetry. Goldberg [10 noti ed thateven the general ase admits a proper tiling of R3.Goldberg's rst family also oin ides with the familyof simpli es found by Hill [14, whose aim was to las-sify re tiable simpli es, that is, simpli es that an be ut by straight uts into nitely many pie es that anbe rearranged to form a ube. The simpli es in Gold-berg's se ond and third families are obtained from thesimpli es in the rst family by splitting into two on-gruent halves. A ording to Sene hal's survey [17, noother spa e-lling tetrahedra than those des ribed bySommerville and Goldberg are known.For general d, Debrunner [5 onstru ted ⌊d/2⌋+ 2one-parameter families and a nite number of addi-tional spe ial types of d-dimensional simpli es thattile Rd. Smith [18 generalized Goldberg's on-stru tion and using Debrunner's ideas, he obtained(⌊d/2⌋ + 2)φ(d)/2 one-parameter families of spa e-lling d-dimensional simpli es; here φ(d) is the Eu-ler's totient fun tion. It is not known whether forsome d there is an a ute spa e-lling simplex or atwo-parameter family of spa e-lling simpli es [18.In re ent years the subje t of tilings has re eiveda ertain impulse from omputer graphi s and other omputer appli ations. In fa t, our original motiva-tion for studying simpli es that are k-reptiles omesfrom a problem of probabilisti marking of Internet

61

On the nonexisten e of k-reptile simpli es in R3 and R4pa kets for IP tra eba k [1, 2. See [15 for a briefsummary of the ideas of this method. For this appli- ation, it would be interesting to nd a d-dimensionalsimplex that is a k-reptile with k as small as possible.For dimension 2 there are several possible typesof k-reptile triangles, and they have been ompletely lassied by Snover et al. [19. In parti ular, k-reptiletriangles exist for all k of the form a2 + b2, a2 or3a2 for arbitrary integers a, b. In ontrast, for d ≥ 3,reptile simpli es seem to be mu h more rare. Theonly known onstru tions of higher-dimensional k-reptile simpli es have k = md. The best known exam-ples are the Hill simpli es (or the HadwigerHill sim-pli es) [5, 12, 14. A d-dimensional Hill simplex is the onvex hull of ve tors 0, b1, b1 + b2, . . . , b1 + · · ·+ bd,where b1, b2, . . . , bd are ve tors of equal length su hthat the angle between every two of them is the sameand lies in the interval (0, π

2+ arcsin 1

d−1).Con erning nonexisten e of k-reptile simpli es indimension d ≥ 3, Hertel [13 proved that a 3-dimensional simplex is an m3-reptile using a stan-dard way of disse tion (whi h we will not dene here)if and only if it is a Hill simplex. He onje turedthat Hill simpli es are the only 3-dimensional reptilesimpli es. Herman Haverkort re ently pointed us toan example of a k-reptile tetrahedron whi h is notHill, whi h ontradi ts Hertel's onje ture. In fa t,ex ept for the one-parameter family of Hill tetrahe-dra, three other spa e-lling tetrahedra des ribed bySommerville [20 and Goldberg [10 are also k-reptilesfor every k = m3. The simpli es and their tiling arebased on the bary entri subdivision of the ube. The onstru tion an be naturally extended to nd simi-lar examples of d-dimensional k-reptile simpli es for

d ≥ 4 and k = md. Matou²ek [15 showed that thereare no 2-reptile simpli es of dimension 3 or larger. Fordimension d = 3 Matou²ek and the se ond author [16proved the following theorem.Theorem 1 [16 In R3, k-reptile simpli es (tetrahe-dra) exist only for k of the form m3, where m is apositive integer.We give a new, simple proof of Theorem 1 in Se -tion 3.Matou²ek and the se ond author [16 onje turedthat a d-dimensional k-reptile simplex an exist onlyfor k of the form md for some positive integer m.We prove a weaker version of this onje ture for four-dimensional simpli es.Theorem 2 Four-dimensional k-reptile simpli es an exist only for k of the form m2, where m is apositive integer.Four-dimensional Hill simpli es are examples of k-reptile simpli es for k = m4. Whether there exists a

four-dimensionalm2-reptile simplex for m non-squareremains an open question.2 Angles in simpli es and Cox-eter diagramsGiven a d-dimensional simplex S with verti esv0, . . . , vd, let Fi be the fa et opposite to vi. A dihe-dral angle βi,j of S is the internal angle of the fa etsFi and Fj , that meet at the (d− 2)-fa e Fi ∩ Fj .An edge-angle of S is the internal (d − 1)-dimensional angle in ident to an edge and an be rep-resented by a (d− 2)-dimensional spheri al simplex.The Coxeter diagram of S is a graph c(S) with la-beled edges su h that the verti es of c(S) representthe fa ets of S and for every pair of fa ets Fi and Fj ,there is an edge ei,j labeled by the dihedral angle βi,j .Observation 3 The edge-angles of a four-dimensional simplex S an be represented byspheri al triangles, whose angles are dihedral anglesin S. Therefore, an edge-angle in S represented by aspheri al triangle with angles α, β, γ orresponds to atriangle in the Coxeter diagram with edges labeled byα, β, γ. The most important tool we use is Debrunner'slemma [5, Lemma 1, whi h onne ts the symmetriesof a d-simplex with the symmetries of its Coxeter di-agram (whi h represents the arrangement of the di-hedral angles). This lemma allows us to substantiallysimplify the proof of Theorem 1 and enables us tostep up by one dimension and prove Theorem 2, whi hseemed unmanageable before.Lemma 4 (Debrunner's lemma [5) Let S be ad-dimensional simplex. The symmetries of S are inone-to-one orresponden e with the symmetries of itsCoxeter diagram c(S) in the following sense: ea hsymmetry ϕ of S indu es a symmetry Φ of c(S) sothat ϕ(vi) = vj ⇔ Φ(Fi) = Fj , and vi e versa.3 A simple proof of Theorem 1We pro eed as in the original proof, but instead of us-ing the theory of s issor ongruen e, Jahnel's theoremabout values of rational angles and Fiedler's theorem,we only use Debrunner's lemma (Lemma 4).Assume for ontradi tion that S is a k-reptile tetra-hedron where k is not a third power of a positive inte-ger. A dihedral angle α is alled indivisible if it annotbe written as a linear ombination of other dihedralangles in S with nonnegative integer oe ients.The following lemmas are proved in [16.

62

XV Spanish Meeting on Computational Geometry, June 26-28, 2013Lemma 5 [16, Lemma 3.1 If α is an indivisible di-hedral angle in S, then the edges of S with dihedralangle α have at least three dierent lengths.Lemma 6 [16, Lemma 3.3 One of the following twopossibilities o ur:(i) All the dihedral angles of S are integer multiplesof the minimal dihedral angle α, whi h has theform π

nfor an integer n ≥ 3.(ii) There are exa tly two distin t dihedral angles β1and β2, ea h of them o urring three times in S.First we ex lude ase (ii) of Lemma 6. If S has twodistin t dihedral angles β1 6= β2, ea h o urring atthree edges, then they an be pla ed in S in two essen-tially dierent ways; see Fig. 1. In both ases, for ea h

i ∈ 1, 2, the Coxeter diagram of S has at least onenontrivial symmetry whi h swaps two distin t edgeswith label βi. By Debrunner's lemma, the orrespond-ing symmetry of S swaps two distin t edges with di-hedral angle βi, whi h thus have the same length. Butthen the edges with dihedral angle βi have at most 2dierent lengths and this ontradi ts Lemma 5, sin ethe smaller of the two angles β1, β2 is indivisible.β1

β1β2

β2 β2

β1

β1

β1β1

β2 β2

β2

Figure 1: Two possible ongurations of two dihedralangles.Now we ex lude ase (i) of Lemma 6. Call the edgesof S (and of c(S)) with dihedral angle α the α-edges.Sin e there are at least three α-edges in S, there is avertex v of S where two α-edges meet. Let β be thedihedral angle of the third edge in ident to v (possiblyβ an be equal to α).We have 2α+ β > π, using a well-known fa t thatthe sum of the three dihedral angles o urring at avertex of S ex eeds π. Writing β = mα = m

n· π, wehave 2 · π

n+ m

n·π > π, whi h implies m > n−2. Sin e

m < n, we have m = n− 1 and hen e β = π − α.Now we distinguish several ases depending on thesubgraph Hα of c(S) formed by the α-edges.• Hα ontains three edges in ident to a ommonvertex (whi h orrespond to a triangle in S).Then all the other edges must have the angle βand we get the onguration as in Fig. 1 (right),whi h we ex luded earlier.• Hα ontains a triangle (the orresponding edgesin S meet at a single vertex). Then β = α, and

thus α = π

2, whi h ontradi ts the ondition n ≥

3 from Lemma 6 (i).• Hα is a path of length three (this orresponds toa path in S, too). Then two edges have the angleβ > α and the remaining edge has some angleγ 6= α. See Figure. 2 (left). The resulting Coxeterdiagram has a nontrivial involution swapping twoα-edges. By Debrunner's lemma, this ontradi tsLemma 5.

• It remains to deal with the ase where Hα is afour- y le (whi h orresponds to a four- y le inS). In this ase the remaining two edges havedihedral angle β, so the Coxeter diagram has adihedral symmetry group D4 a ting transitivelyon the α-edges. By Debrunner's lemma, all theα-edges have the same length. This again on-tradi ts Lemma 5.

α

αβ

γ β

α

α

αβ

α β

α

Figure 2: The α-edges form a path (left) or a four- y le (right) in c(S)We obtained a ontradi tion in ea h of the ases,hen e the proof of Theorem 1 is nished.4 The proof of Theorem 2The method of the proof is similar to the three-dimensional ase.Assume for ontradi tion that S is a four-dimensional k-reptile simplex where k is not a squareof a positive integer. Let S1, . . . , Sk be mutually on-gruent simpli es similar to S that tile S. Then ea hSi has volume k-times smaller than S, and thus Si iss aled by the ratio ρ := k−1/4 ompared to S. Fork non-square, ρ is an irrational number of algebrai degree 4 over Q.Similarly to [16 we dene an indivisible edge-angle(spheri al triangle) as a spheri al triangle whi h an-not be tiled with smaller spheri al triangles represent-ing the other edge-angles of S or their mirror images.Clearly, the edge-angle with the smallest surfa e areais indivisible. We onsider a spheri al triangle and itsmirror image as the same spheri al triangle.We obtain a result similar to Lemma 5: if T0 is anindivisible edge-angle in S, then the edges of S withedge-angle T0 have at least four dierent lengths (andin parti ular, there are at least four su h edges).

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On the nonexisten e of k-reptile simpli es in R3 and R4The strategy of the proof is now the following. Firstwe ex lude the ase of two indivisible edge-angles, us-ing only elementary ombinatorial arguments and De-brunner's lemma.Then we onsider the ase of one indivisible edge-angle. Here we need more involved arguments. Westudy the problem of tiling spheri al triangles by on-gruent triangular tiles, whi h might be of independentinterest. We give a partial lassi ation of su h tilings,whi h might be possible to extend to a full lassi a-tion using a reasonable amount of eort. A relatedquestion, a lassi ation of edge-to-edge tilings of thesphere by ongruent triangles, has been ompletelysolved by Agaoka and Ueno [3.To rule out several ases, we use a hara teriza-tion of (d+1

2

)-tuples of dihedral angles by Fiedler [7.The hara terization implies, in parti ular, that ifβi,j , i, j = 1, 2, . . . , d + 1, are the dihedral an-gles of some d-dimensional simplex, then the ma-trix (ai,j ; i, j = 1, 2, . . . , n + 1), where ai,i = 0 andai,j = cosβi,j for i 6= j, is singular.Referen es[1 M. Adler, Tradeos in probabilisti pa ket mark-ing for IP tra eba k, Pro . 34th Annu. ACMSymposium on Theory of Computing (2002) 407418.[2 M. Adler, J. Edmonds and J. Matou²ek, Towardsasymptoti optimality in probabilisti pa ketmarking, Pro . 37th Annu. ACM Symposium onTheory of Computing (2005) 450459.[3 Y. Agaoka and Y. Ueno, Classi ation of tilingsof the 2-dimensional sphere by ongruent trian-gles, Hiroshima Math. J. 32(3) (2002), 463540.[4 C. Bandt, Self-similar sets. V. Integer matri esand fra tal tilings of Rn, Pro . Amer. Math. So .112(2) (1991), 549562.[5 H. E. Debrunner, Tiling Eu lidean d-spa e with ongruent simplexes, Dis rete geometry and on-vexity (New York, 1982), vol. 440 of Ann. NewYork A ad. S i., New York A ad. S i., New York(1985) 230261.[6 A. L. Edmonds, Sommerville's missing tetrahe-dra, Dis rete Comput. Geom. 37(2) (2007), 287296.[7 M. Fiedler, Geometry of the simplex in En (inCze h, with English and Russian summary), a-sopis pro p¥stování matematiky 79 (1954), 297320.[8 M. Gardner, The unexpe ted hanging and othermathemati al diversions , The University ofChi ago Press (1991).

[9 G. Gelbri h, Crystallographi reptiles, Geom.Dedi ata 51(3) (1994), 235256.[10 M. Goldberg, Three innite families of tetrahe-dral spa e-llers, J. Comb. Theory, Ser. A 16(1974), 348354.[11 S. W. Golomb, Repli ating gures in the plane,Mathemati al Gazette 48 (1964), 403412.[12 H. Hadwiger, Hills he Hypertetraeder (in Ger-man), Gaz. Mat. (Lisboa) 12(50) (1951), 4748.[13 E. Hertel, Self-similar simpli es, Beiträge AlgebraGeom. 41(2) (2000), 589595.[14 M. J. M. Hill, Determination of the volumesof ertain spe ies of tetrahedra without employ-ment of the method of limits, Pro . LondonMath. So . 27 (1896), 3952.[15 J. Matou²ek, Nonexisten e of 2-reptile sim-pli es, Dis rete and Computational Geometry:Japanese Conferen e, JCDCG 2004 , Le tureNotes in Computer S ien e 3742, Springer, Berlin(2005) 151160, erratum at http://kam.mff. uni. z/~matousek/no2r-err.pdf.[16 J. Matou²ek and Z. Safernová, On the nonexis-ten e of k-reptile tetrahedra, Dis rete Comput.Geom. 46(3) (2011), 599609.[17 M. Sene hal, Whi h tetrahedra ll spa e?, Math.Magazine 54(5) (1981), 227243.[18 W. D. Smith, Pythagorean triples, ratio-nal angles, and spa e-lling simpli es (2003),manus ript, http:// iteseerx.ist.psu.edu/viewdo /summary?doi=10.1.1.124.6579.[19 S. L. Snover, C. Waiveris and J. K. Williams,Rep-tiling for triangles, Dis rete Math. 91(2)(1991), 193200.[20 D. M. Y. Sommerville, Division of spa e by on-gruent triangles and tetrahedra, Pro . Roy. So .Edinburgh 4 (1923), 85116.

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http://kam.mff.cuni.cz/~matousek/no2r-err.pdf

http://kam.mff.cuni.cz/~matousek/no2r-err.pdf

http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.124.6579

http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.124.6579



S. Bereg∗1, R. Fabila-Monroy†2, D. Flores-Peñaloza‡3, M.A. Lopez§4, and P. Pérez-Lantero¶5

1Department of Computer Science, University of Texas at Dallas, USA.2CINVESTAV, Instituto Politécnico Nacional, Mexico.

3Departamento de Matemáticas, Facultad de Ciencias, UNAM, Mexico.4Department of Computer Science, University of Denver, USA.

5Escuela de Ingeniería Civil en Informática, Universidad de Valparaíso, Chile.

Abstract

In 1926, Jarník introduced the problem of drawing aconvex n-gon with vertices having integer coordinates.He constructed such a drawing in the grid [1, c ·n3/2]2

for some constant c > 0, and showed that this gridsize is optimal up to a constant factor. We considerthe analogous problem of drawing the double circle,and prove that it can be done within the same gridsize. Moreover, we give an O(n log n)-time algorithmto construct such a point set.

1 Introduction

Given n ≥ 3, a double circle is a set P =p0, p1, . . . , pn−1, p′0, p′1, . . . , p′n−1 of 2n planar pointsin general position such that: (1) p0, p1, . . . , pn−1 areprecisely the vertices of the convex hull of P labelledin counterclokwise order around the boundary; (2)point p′i is close to the segment joining pi with pi+1;(3) the line passing through pi and p′i separates pi+1

from P ; and (4) the line passing through p′i and pi+1

separates pi from P (see Figure 1). Subindices aretaken modulo n. The double circle has been consid-ered in combinatorial geometry and it is conjecturedto have the least number of triangulations [1, 2].Drawing an n-vertex convex polygon with integer

vertices can be easily done by considering the n points(1, 1), (2, 4), (3, 9), . . . , (n, n2) as the vertices of thepolygon. In this case the size of the integer pointset is equal to n2 − 1 = Θ(n2), where size refers tothe smallest N such that the point set can be trans-lated to lie in the grid [0, N ]2. In 1926, Jarník [5]showed how to draw an n-vertex convex polygon with

∗Email: [email protected].†Email: [email protected]. Partially sup-

ported by grant 153984 (CONACyT, Mexico).‡Email: [email protected]. Partially supported by

grants 168277 (CONACyT, Mexico) and IA102513 (PAPIIT,UNAM, Mexico).§Email: [email protected].¶Email: [email protected]. Partially supported by grant

CONICYT, FONDECYT/Iniciación 11110069 (Chile).

p0p1

p2

p3 p4

p5

p′0

p′1

p′2p′3

p′4

p′5

Figure 1: A double circle of twelve points.

size N = O(n3/2) and proved that this bound is op-timal. In recent years the so-called Jarník polygonsand extensions of them have been studied [3, 6].Given any integer point (i, j), we say that (i, j)

is visible (from the origin) if the interior of the linesegment joining the origin and (i, j) contains no lat-tice points. Observe that (i, j) is visible if and onlyif gcd(i, j) = 1, where gcd(i, j) denotes the great-est common divisor of i and j. We consider pointsas vectors as well, and vice versa. A Jarník poly-gon is formed by choosing a natural number Q, andtaking the set VQ of visible vectors (i, j) such thatmax|i|, |j| ≤ Q [4, 5, 6]. The polygon is thenthe unique (up to translation) convex polygon whoseedges, viewed as vectors, are precisely the elementsof VQ, that is, the vertices can be obtained by start-ing from an arbitrary point and adding the vectorsof VQ, one by one, in counterclockwise order, to thepreviously computed vertex (see Figure 2).

V2

Figure 2: A Jarník polygon (right) and its generatingvectors V2 (left).

We study how to draw a 2n-point double circle withinteger points using the smallest size N . We present

65


anO(n log n)-time algorithm that correctly constructsthe double circle with size within O(n3/2), where thatbound is also optimal. In Section 2 we show our al-gorithm, and in Section 3 its correctness is proved.Finally, in Section 4, we state future work.

2 Double circle construction

Observe that a simple construction with quadraticsize is as follows: Consider the function f(x) = x2+x.For i = 1, . . . , 2n − 1, add the point (i, f(i)) if i isodd, and the point (i, f(i) + 2) otherwise. The finalpoint is (n, f(2n−1)+f(1)

2 − 1) = (n, 2n2 − n), i.e., thepoint just below the midpoint of the segment con-necting (1, f(1)) and (2n− 1, f(2n− 1)). The size ofthe resulting point set is N = f(2n − 1) − f(1) =(2n− 1)2 + (2n− 1)− 2 = 4n2 − 2n− 2 = Θ(n2).We say that a sequence V of vectors is symmetric

if V contains an even number of vectors sorted coun-terclockwise around the origin, and for every vector ain V its opposite vector −a is also in V . Observe thatany sequence of vectors defining a Jarník polygon issymmetric. For any sequence V = [v1, v2, . . . , v2t] of2t vectors let the point set P(V ) := p1, p2, . . . , p2t,where p1 = v1 and pi = pi−1 + vi for i = 2, . . . , 2t.Note that if we sort the elements of VQ around theorigin then the elements of P(VQ) are the vertices ofthe Jarník polygon. Furthermore, if V is symmetricthen the elements of P(V ) are in convex position. Letsequence alt(V ) := [v2, v1, v4, v3, . . . , v2t, v2t−1] (seeFigure 3 for an example with t = 8). For any scalarλ let the sequence λV := [λv1, λv2, . . . , λv2t].

Figure 3: P(alt(V4)).

The idea is to generate a suitable symmet-ric sequence V of 2n vectors and then buildthe point set P(alt(V )) as the double cir-cle point set, up to some transformation ofthe elements of alt(V ). A (not optimal) ex-ample is V = [(1, 1), (1, 2), . . . , (1, n), (−1,−1),(−2,−2), . . . , (−1, n)] for even n ≥ 4. The point setP(alt(V )) is in fact a double circle but its size is equalto 1 + 2 + . . .+ n = Θ(n2) (see Figure 4).The construction in which the resulting point set is

a double circle of size O(n3/2) is based on the next

Figure 4: A naive construction for n = 4 showing bothvectors (left) and the resulting point set (right).

two algorithms:VisibleVectors(n): With input n ≥ 3, the sym-

metric sequence V of 2n visible vectors, sorted coun-terclockwise around the origin, is generated so as tosatisfy the next two invariants. Let Bt := p ∈ Z2 :‖p‖1 ≤ t, k := maxv∈V ‖v‖1, and (even) s be thenumber of visible vectors of Bk−1: (i) all visible vec-tors of Bk−1 are in V , and (ii) the other elements of Vare generated as follows, until 2n− s elements are ob-tained: for i = 1, . . . , k− 1 generate vectors (i, k− i),(−i,−(k− i)), (−i, k− i), (i,−(k− i)) in this order, ifand only if gcd(i, k − i) = 1. Refer to Algorithm 2.1for a pseudo-code.

BuildDoubleCircle(n): With input n ≥ 3,build a 2n-point double circle. First, set sequencesV :=VisibleVectors(n) and [v′1, v

′2, . . . , v

′2n] :=

alt(V ). Then, the sequence W = [w1, w2, . . . , w2n] of2n vectors is created as follows: for i = 1, 3, . . . , 2n−1set wi = (1−λ)v′i+λv

′i+1 and wi+1 = λv′i+(1−λ)v′i+1,

where λ = 1/3. Finally, build the 2n-point setP((1/λ)W ) as the double circle.

Algorithm 2.1: VisibleVectors(n)

k ← 1, V ← [(1, 0), (−1, 0), (0, 1), (0,−1)]repeatk ← k + 1for i← 1 to k − 1

do

j ← k − iif gcd(i, j) = 1

then

if length(V ) < 2nthen V ← V + [(i, j), (−i,−j)]

if length(V ) < 2nthen V ← V + [(−i, j), (i,−j)]

until length(V ) = 2nSort V counterclockwise around originreturn (V )

3 Construction correctness

Let V = [v1, v2, . . . , v2n] be the (circular) sequence ofvectors obtained by executing VisibleVectors(n),

66


for n ≥ 3. For every i = 1, 3, 5, . . . , 2n−1 we say thatthe pair of vectors vi, vi+1 is a pair of alt(V ).

Lemma 1 (Chapter 2 of [4]) Given a naturalnumber Q, the number |VQ| of vertices of the Jarníkpolygon is equal to

4 + 4

Q∑i=1

Q∑j=1

1

gcd(i,j)=1

=24Q2

π2+O(Q logQ).

The size S(Q) of the Jarník polygon is equal to

1 + 2

Q∑i=1

Q∑j=1

i

gcd(i,j)=1

= 1 + 2

Q∑i=1

Q∑j=1

j

gcd(i,j)=1

=6Q3

π2+O(Q2 logQ).

Lemma 2 V is symmetric and point set P(V ) hassize O(n3/2).

Proof. Observe that for every vector a in V , −a isalso in V since in algorithm VisibleVectors the vec-tors are added to sequence V in pairs, and each pairconsists of two opposite vectors. Then V becomessymmetric once the elements of V are sorted coun-terclockwise around the origin. On the other handVb k−1

2 c⊂ V ⊂ Vk, where k = maxv∈V ‖v‖1. Then we

have |Vb k−12 c| ≤ 2n ≤ |Vk|, which implies k = Θ(

√n)

by Lemma 1. By the same lemma we obtain:

n∑i=1

x(vi),

n∑i=1

y(vi) < 1 + 2

k∑i=1

k∑j=1

i

gcd(i,j)=1

= S(k)

= Θ(k3) = Θ(n3/2)

Hence, the size of P(V ) is O(n3/2).

Let o denote the origin of coordinates. Given twopoints p, q let `(p, q) denote the line passing through pand q and directed from p to q, and pq denote the seg-ment joining p and q. Given three points p = (xp, yp),q = (xq, yq), and r = (xr, yr), let ∆(p, q, r) denotethe triangle with vertices at p, q, and r; A(p, q, r) de-note de area of ∆(p, q, r); and turn(p, q, r) denote theso-called geometric turn (going from p to r passingthrough q) where

turn(p, q, r) =

∣∣∣∣∣∣xp yp 1xq yq 1xr yr 1

∣∣∣∣∣∣and A(p, q, r) = 1

2 |turn(p, q, r)|. Extending this no-tation, let ∆(p, q) := ∆(o, p, q), A(p, q) := A(o, p, q),and turn(p, q) := turn(o, p, q). We use the so-calledPick’s theorem:

Theorem 3 (Pick’s theorem [7]) The area of anysimple polygon H with lattice vertices is equal to i +b/2−1, where i and b are the numbers of lattice pointsin the interior and the boundary of H, respectively.

Lemma 4 For every two consecutive vectors a1, a2 ofV we have A(a1, a2) = 1/2.

Proof. Suppose ∆(a1, a2) contains a lattice point pdifferent from o, a1, and a2. Then p cannot belongto segments oa1 and oa2, and segment op containsa visible point q (possibly equal to p). If ‖q‖1 <max‖a1‖1, ‖a2‖1 then q must belong to V by in-variant (i) of algorithm VisibleVectors. Other-wise, we have ‖q‖1 = max‖a1‖1, ‖a2‖1. Supposew.l.o.g. that ‖a1‖1 < ‖a2‖1, and let the point q′ de-note the intersection of `(o, q) with the segment s con-necting a1 to a2. Observe that q′ = δa1 + (1 − δ)a2for some δ ∈ (0, 1), and further that ‖q‖1 ≤ ‖q′‖1 =‖δa1 + (1− δ)a2‖1 ≤ δ‖a1‖1 + (1− δ)‖a2‖2 < ‖a2‖1,which is a contradiction. Then we must have that‖q‖1 = ‖a1‖1 = ‖a2‖1, which implies that q, a1, a2 be-long to a same quadrant since in this case q is at theinterior of the segment s. Therefore, q must belong toV by invariant (ii) of algorithm VisibleVectors. Inboth cases, the fact that q belongs to V contradictsthe fact that a1 and a2 are consecutive vectors of V .Hence A(a1, a2) = 1/2 by Pick’s theorem.

Let λ ∈ (0, 1/2). Given a pair a, b of vectors leth(λ, a, b) := (1− λ)a+ λb (see Figure 5).

a

b

h(λ, a, b)

h(λ, b, a)

Figure 5: Two vectors a and b, and the vectors h(λ, a, b)and h(λ, b, a).

Lemma 5 Let a1, a2, a3, a4 be four consecutive vec-tors of V such that a1, a2 and a3, a4 are pairs ofalt(V ). Let λ ∈ (0, 1/2), q1 = h(λ, a2, a1), q2 =q1 + h(λ, a1, a2), q3 = q2 + h(λ, a4, a3), and q4 =q3 + h(λ, a3, a4). Then q2 is to the right of `(o, q1)and both q3 and q4 are to the left of `(o, q1).

Proof. (Refer to Figure 6.) We have turn(q1, q2) =turn(q1, q1 +h(λ, a1, a2)) = turn((1−λ)a2 +λa1, a1 +a2) = (1 − λ) turn(a2, a1) + λ turn(a1, a2) = 2(2λ −1)A(a1, a2) < 0, which implies that q2 is to the rightof the line `(o, q1). On the other hand:

turn(q1, q3)

= turn(h(λ, a2, a1), h(λ, a2, a1) + h(λ, a1, a2) +

h(λ, a4, a3))

= turn(h(λ, a2, a1), h(λ, a1, a2)) +

turn(h(λ, a2, a1), h(λ, a4, a3))

67


= turn((1− λ)a2 + λa1, (1− λ)a1 + λa2)) +

turn((1− λ)a2 + λa1, (1− λ)a4 + λa3))

= (1− λ)2 turn(a2, a1) + λ2 turn(a1, a2) +

(1− λ)2 turn(a2, a4) + λ(1− λ) turn(a2, a3) +

λ(1− λ) turn(a1, a4) + λ2 turn(a1, a3)

= 2(

(2λ− 1)A(a1, a2) + (1− λ)2A(a2, a4) +

λ(1− λ)A(a2, a3) + λ(1− λ)A(a1, a4) +

λ2A(a1, a3))

= 2

(1

2(2λ− 1) + (1− λ)2A(a2, a4) + (1)

λ(1− λ)A(a2, a3) + λ(1− λ)A(a1, a4) +

λ2A(a1, a3)

)≥ (2λ− 1) + (1− λ)2 + λ(1− λ) + (2)

λ(1− λ) + λ2

= 2λ > 0

where equation (1) follows from Lemma 4 and equa-tion (2) follows from the fact that by Pick’s the-orem the area of any non-empty triangle with lat-tice vertices is at least 1/2. Therefore, q3 isto the left of `(o, q1). Similarly, since we havethat turn(ai, aj) > 0 (i = 1, 2; j = 3, 4) thenturn(h(λ, a2, a1), h(λ, a3, a4)) > 0, which implies thatq4 is to the left of `(o, q1) given that q3 is to theleft of `(o, q1). By symmetry, it can be proved thatturn(q4, q3, q1) < 0 and turn(q4, q3, o) < 0, implyingthat both q1 and o are to the right of `(q4, q3).

o

q1

q2

q3

q4

`(o, q1)

h(λ, a2, a1)

h(λ, a1, a2)

h(λ, a4, a3)

h(λ, a3, a4)

Figure 6: Proof of Lemma 5.

Theorem 6 There is an O(n log n)-time algorithmthat for all n ≥ 3 builds a double circle of 2n pointsin the grid [0, N ]2 where N = O(n3/2).

Proof. Execute the algorithm BuildDoubleCir-cle with input n, being V the result of callingVisiblePoints(n), building the point set P of 2npoints. Observe that λ = 1/3 implies that point

wi/λ = 3wi is integer for i = 1 . . . 2n, and thenall elements of P are integer points. By Lemma 5point set P is a double circle. The size of P(V ) isO(n3/2) by Lemma 2, and since all elements of P be-long to the polygon with vertices P(3V ) the size Nof P is also O(n3/2). Finally, translate P to lie in thegrid [0, N ]2. In algorithm VisiblePoints the timecomplexity is dominated by: (1) computing gcd(i, j)for O((

√n)2) = O(n) pairs i, j; and (2) sorting vec-

tors V counterclockwise around the origin. In Case(1) the time complexity is O(n log n) since gcd(i, j)consumes O(log(mini, j)) = O(log

√n) = O(log n)

time. Case (2) consumes O(n log n) time as well.Since the time complexity of VisiblePoints dom-inates the time complexity of the main algorithmBuildDoubleCircle, the result follows.

4 Future workWe are working on extending these results to buildother known point sets in integer points of small size,such as the double convex chain, the Horton set, andothers. We plan to eventually release a software li-brary supporting many of these constructions.

AcknowledgementsThe problem studied here were introduced and partiallysolved during a visit to University of Valparaiso funded byproject CONICYT Fondecyt/Iniciación 11110069 (Chile).The authors would like to thank anonymous referees fortheir valuable comments.

References[1] O. Aichholzer, F. Hurtado, and M. Noy. A lower bound

on the number of triangulations of planar point sets.Computational Geometry, 29(2):135–145, 2004.

[2] O. Aichholzer, D. Orden, F. Santos, and B. Speck-mann. On the number of pseudo-triangulations of cer-tain point sets. Journal of Combinatorial Theory, Se-ries A, 115(2):254–278, 2008.

[3] I. Bárány and N. Enriquez. Jarník’s convex lat-tice n-gon for non-symmetric norms. MathematischeZeitschrift, 270:627–643, 2012.

[4] M. N. Huxley. Area, lattice points, and exponentialsums. Oxford Science Publications. The ClarendonPress Oxford University Press, New York, 1996.

[5] V. Jarník. Über die Gitterpunkte auf konvexen Kur-ven. Mathematische Zeitschrift, 24:500–518, 1926.

[6] G. Martin. The Limiting Curve of Jarník’s Polygons.Transactions of the American Mathematical Society,355(12):4865–4880, 2003.

[7] G. Pick. Geometrisches zur Zahlenlehre. Sitzungs-berichte des Deutschen Naturwissenschaftlich-Medicinischen Vereines für Böhmen "Lotos" in Prag.,19:311–319, 1899.

68



David N. de Miguel∗1, David Orden†2, Encarnación Fernández-Fernández‡3, José M. Rodríguez-Nogales §3, andJosena Vila-Crespo ¶4

1Universidad de Alcalá, Spain.2Departamento de Física y Matemáticas, Universidad de Alcalá, Spain.

3Departamento de Ingeniería Agrícola y Forestal, Universidad de Valladolid, Spain.4Departamento de Anatomía Patológica, Medicina Preventiva y Salud Pública, Medicina Legal y Forense,

Universidad de Valladolid, Spain.

Abstract

Sensory evaluation of foods is as important as chem-ical, physical or microbiological examinations, beingspecially relevant in food industries. Classical meth-ods can be long and costly, making them less suitablefor certain industries like the wine industry. Some al-ternatives have arisen recently, including Nappingr,where the tasters represent the sensory distances be-tween products by positioning them on a tablecloth;the more similar they perceive the products, the closerthey position them on the tablecloth. This methoduses multiple factor analysis (MFA) to process thedata collected. The present paper introduces thesoftware SensoGraph, which makes use of proximitygraphs to analyze those data. The application is de-scribed and experimental results are presented in or-der to compare the performances of SensoGraph andNappingr, using eight wines from the Toro region andtwo groups of twelve tasters with dierent expertise.

1 Sensory analysis

The goal of sensory evaluation of foods is the studyof the sensations they produce. When consuming afood, stimulus of several classes are perceived; vi-sual (color, shape, brightness), tactile (at ngertipsor mouth epithelium), odorous (at nose epithelium),gustatory (at taste buds), and even auditory (e.g., forcrunchy food). The norm ISO 5492:2008 denes sen-sory analysis as the science related to the evaluationof organoleptic attributes of a product by the senses,and other ISO norms unify the tools and methods

∗Email: [email protected].†Email: [email protected]. Research partially supported

by MICINN Project MTM2011-22792, ESF EUROCORES

programme EuroGIGA - ComPoSe IP04 - MICINN Project

EUI-EURC-2011-4306 and Junta de Castilla y León Project

VA172A12-2.‡Email: [email protected].§Email: [email protected].¶Email: [email protected].

used for that evaluation. The sensory examinationturns out to be as important as chemical, physical ormicrobiological examinations, being specially relevantin food industries.The main tool in sensory analysis is a panel of

tasters, either experts or consumers, who evaluatethe products from an analytic and/or hedonic pointof view. As any other instrument depends on itscalibration, such a panel depends on human be-ings. Their perceptions are translated into quanti-able data, which is then treated by means of dierentmethods.The classical method is descriptive analysis, which

aims to describe the sensory characteristics of a prod-uct and use them to quantify the sensory dierencesbetween products [11]. Dierent implementations ofthis method provide a quantitative description of thesensory attributes perceived by a group of experttasters, chosen because of their sensory abilities, whoare trained to describe and evaluate sensory dier-ences among products. Such a training is a criticalstep in the creation of an expert panel of tasters, whentasters agree on the denitions of descriptors and theuse of scales, in order to provide reliable and consis-tent results.However, this training can be long and costly, mak-

ing it less suitable for certain industries like the wineindustry. There, sensory characterization is usuallyperformed by the oenologist in charge of the winery,for whom it is dicult to enrol in a panel requiring aregular activity during a long time. Thus, in the lastyears several alternative methods have been proposed,aiming to provide a fast sensory positioning of a set ofproducts, in order to avoid the most time-consumingsteps in classical methods. A prominent one amongthese alternatives is Nappingr [12]. In a single ses-sion, all the products are provided simultaneously tothe tasters, who represent the sensory distances be-tween products by positioning them on a tablecloth.Products which are perceived as similar should be po-sitioned close to each other, while products perceivedas dierent should be positioned far enough. Each

69


taster chooses the criteria and the relative importancegiven. These data are later processed using multiplefactor analysis (MFA) [4], in order to take into ac-count the criteria and relative importance of all thetasters. Despite having both advantages and disad-vantages, Nappingr has become a useful tool whensome accuracy can be sacriced for the sake of a fasterstudy [13].

2 Proximity graphs

Given a set of points in the plane, a (geometric)proximity graph connects two of them according toa chosen proximity criterion. These graphs have beenwidely used in order to analyze the relative positionof points, for instance looking for clusters or span-ning structures. See [1, 7] and the references therein.Thus, it seems natural to use them in order to an-alyze the data collected by a tablecloth method forsensory analysis, like Nappingr. Among the manydierent types of proximity graphs, we have chosenthe following:

Nearest Neighbor Graph (NNG): Each point isjoined to the closest among the remaining points [14].

k-Nearest Neighbor Graph (k-NNG): In this gen-eralization of the NNG, each point is joined to the kclosest among the remaining points.

Minimum Spanning Tree (MST): Among thetrees passing through all the given points, the MST isthe one which minimizes the sum of edge lenghts [10].

Relative Neighborhood Graph (RNG): Thisgraph, introduced by Toussaint [15], joins two pointsP,Q if there is no point whose distances to both Pand Q are smaller than the distance d(P,Q).

k-Relative Neighborhood Graph (k-RNG): Thegeneralization of the RNG which allows up to k pointswith distances to both P and Q smaller than d(P,Q).

Gabriel Graph (GG): In this graph two points P,Qare joined if there is no other point inside the closeddisk which has the segment PQ as diameter [5].

k-Gabriel Graph (k-GG): Generalization of the GGallowing up to k points to lie inside the closed disk.

Delaunay Triangulation (DT): Three pointsP,Q,R form a triangle precisely if their circumcircledoes not contain any other point [3].

k-Delaunay Triangulation (k-DT):Generalizationof the DT allowing up to k points to lie inside theclosed disk.

β-skeleton (β-SK): A family of proximity graphs,one for each value of β ≥ 0, see [9] for more details.For β = 1 we get the GG. For β = 2 we get the RNG.

Unit Disk Graph (UDG): In this graph two pointsP,Q are joined if the distance d(P,Q) between themis no greater than a xed threshold [2].

3 The SensoGraph application

SensoGraph is an application, still under develop-ment, which aims to use proximity graphs for the anal-ysis of data collected from tablecloth sensory meth-ods like Nappingr. The interface intends to be intu-itive and easy to use, so that no special knowledge isneeded.

Tasting data are stored in the form of an m × 2nmatrix, in which there is a row per product and a pairof columns per taster, storing the two coordinates as-signed to the corresponding product. In a rst screen,this matrix can be manually created or inserted froma CSV le, according to the IETF RFC 4180 stan-dard. After insertion, the matrix can be modied byadding or removing rows or columns, as well as byediting an individual entry. See Figure 1.

Figure 1: Matrix of tasting data.

After accepting the data matrix, a new screen isshown. There, the user can choose a type of proxim-ity graph among the ones specied in Section 2. Forthose proximity graphs depending on a parameter, k-NNG, k-RNG, k-GG, k-DT, β-SK, and UDG, the usercan change the value of the parameter. Furthermore,the application allows to intersect any of the graphsconsidered with the UDG, in case the user wants toavoid too long edges.

For the given choice of a type of proximity graph,the application generates the graph for each of thetaster's tablecloths. The user can choose a taster andcheck its tablecloth and the resulting graph. Whenvisualizing a tablecloth, it is also possible to changethe type of proximity graph, in order to check thedierences between them. See Figure 2.

Figure 2: Tablecloth for taster number 2 with theUDG for radius 10.

70


Furthermore, chosen a type of proximity graph, theuser can also merge all the tablecloths and their corre-sponding graphs into a single, global, picture. There,every edge is shown with a thickness, computed ac-cording to a simple function: The thickness corre-sponds to the number of tasters for which that edgeappears in the proximity graph. This representationencodes the global opinion of the panel of tasters, sothat those products perceived as similar are joined bythicker edges, while products considered dierent arejoined by thinner edges (or even not joined at all).

In order to position the vertices of this global pic-ture, SensoGraph considers the edges like springswhich try to approach their endpoints, with a strengthproportional to the edge thickness. Thus, verticesjoined by thicker edges become closer than thosejoined by thinner edges. In this new picture, the prod-ucts considered similar by the panel of tasters can berecognized not only by the thickness of the edge be-tween them, but also by their mutual distance, seeFigures 4, 6, and 8. Such a positioning is performedby a slight adaptation of the algorithm by Kamadaand Kawai [8].

Since some types of proximity graphs insist in join-ing vertices which are far apart, SensoGraph oers asecond way of computing the thickness of an edge.The distance function also looks only at the edgesappearing in the proximity graph chosen but, in addi-tion, it takes into account their length, decreasing thecontribution of long edges to the total thickness. Asmentioned above, the user can also choose to discardtoo long edges, by intersecting any of the proximitygraphs with the UDG.

Furthermore, the user can also peel the graph in theglobal picture, by removing edges below any chosenthickness, in order to keep only the most relevant ones.

4 Experimental results

Eight wines from the Toro region, elaborated us-ing dierent yeasts during the alcoholic fermentation,were considered. Two panels, of twelve non-trainedtasters each, were selected. The experts panel wascomposed by people, mainly young, with some knowl-edge of the techniques for sensory analysis of wines.The non-experts panel was composed by plain con-sumers, with dierent ages and levels of knowledge.Each of the panels performed a session of Nappingr,and the experts panel repeated for a second session, inorder to check for improvements due to such a slighttraining.

The data from those three sessions was then pro-cessed both by multiple factor analysis (MFA) [6], asusual in Nappingr, and by SensoGraph. Figures 3to 8 show the results obtained, with SensoGraph us-ing GG and the simple thickness function.

Figure 3: Experts panel, Nappingr.

Figure 4: Experts panel, SensoGraph.

Figure 5: Repetition of experts panel, Nappingr.

Figure 6: Repetition of experts panel, SensoGraph.

71


Figure 7: Non-experts panel, Nappingr.

Figure 8: Non-experts panel, SensoGraph.

Among the three sessions performed, the most rep-resentative results of Nappingr were those from therepetition of the experts panel. Figure 5 shows a35.25% of the variation explained by the rst di-mension and a 26.23% of the remaining variation ex-plained by the second dimension, for a total inertia of61.48%. Being this the session for which Nappingr

is most reliable, it is the one chosen to compare withthe results provided by SensoGraph.

For that session, using SensoGraph with the sim-

ple thickness function provides the same clusters asNappingr for all the graphs considered except forNNG, which has too few edges, and for k-DT, whichhas too many edges. Using the distance thicknessfunction does not change the elements in the clus-ters, although the whole picture appears expandedand highlights only the strongest connections betweendierent clusters. Furthermore, it improves the re-sults for the extreme cases above, leading to the sameclusters as Nappingr for k-DT and palliating the dif-ferences for NNG.

A possible advantage of SensoGraph is giving otherkind of information than Nappingr. Apart from pro-viding several types of proximity graphs and parame-ters to test with, SensoGraph shows how the dierentclusters are connected. For an example, Figures 5and 6 lead to the same three clusters, but only theone from SensoGraph shows that they are actuallyquite connected, reecting the fact that all the winesconsidered were actually quite similar [6].

5 Acknowledgements

The authors want to gratefully thank Ferran Hurtadofor proposing them the use of proximity graphs inorder to analyze data from tablecloth methods.

References

[1] J. Cardinal, S. Collette, and S. Langerman, Emptyregion graphs, Computational Geometry: Theory andApplications, 42 (2009), 183195.

[2] B. N. Clark, C. J. Colbourn, and D. S. Johnson, UnitDisk Graphs, Discrete Mathematics, 86 (1990), 165177.

[3] B. N. Delaunay, Sur la sphere vide, Bulletin of the

Academy of Sciences of the USSR, VII (1934), 793800.

[4] B. Escoer and J. Pagès, Analyses factorielles simpleset multiples, Dunod, Paris, 1998.

[5] K. R. Gabriel and R. R. Sokal, A new statistical ap-proach to geographic variation analysis, SystematicBiology, 18:3 (1969), 259278.

[6] L. Gallego-Expósito, Nuevas técnicas de análisis sen-sorial de alimentos: Métodos espaciales, Degree the-sis, 2011.

[7] J.W. Jaromczyk and G.T. Toussaint, Relative neigh-borhood graphs and their relatives, Proceedings of theIEEE, 80:9 (1992), 15021517.

[8] T. Kamada and S. Kawai, An algorithm for draw-ing general undirected graphs, Information Process-

ing Letters, 31 (1989), 715.

[9] D. Kirkpatrick and J. Radke, A framework for com-

putational morphology, in: Computational Geome-try, Machine Intelligence and Pattern Recognition,2, North-Holland, Amsterdam, 1985, 217248.

[10] J. B. Kruskal, On the shortest spanning subtree of agraph and the traveling salesman problem, Problemsof the American Mathematical Society, 7 (1956), 4850.

[11] H. T. Lawless and H. Heymann, Principles of SensoryEvaluation, Aspen Publishers, Gaithersburg, 1999.

[12] J. Pagès, Collection and analysis of perceived productinter-distances using multiple factor analysis: Appli-cation to the study of 10 white wines from the LoireValley, Food Quality and Preference, 16 (2005), 642649.

[13] L. Perrin , R. Symoneaux, I. Maître, C. Asselin, F.Jourjon , and J. Pagès, Comparison of three sensorymethods for use with the Napping procedure: Caseof ten wines from Loire valley, Food Quality and Pref-erence, 19 (2008), 111.

[14] M. I. Shamos and D. Hoey, Closest-point problems,in Proceedings of FOCS, 1975, 151162.

[15] G. T. Toussaint, The relative neighborhood graph ofa nite planar set, Pattern Recognition, 12 (1980),261268.

72


Simulated Annealing applied to the MWPT problem

E.O. Gagliardi∗1, M. G. Leguizamón †1, and G. Hernández ‡2

1Universidad Nacional de San Luis, Argentina2Universidad Politécnica de Madrid, España

Abstract

The Minimum Weight Pseudo-Triangulation(MWPT) problem is suspected to be NP-hard.We show here how Simulated Annealing (SA) canbe applied for obtaining approximate solutions tothe optimal ones. To do that, we applied two SAalgorithms, the basic version and our extended hybridversion of SA. Through the experimental evaluationand statistical study we assess the applicability andperformance of the SA algorithms. The obtainedresults show the benets of using the hybrid version ofSA to achieve improved and higher quality solutionsfor the MWPT problem.

Introduction

A pseudo-triangle is a simple polygon with three con-vex vertices, and a pseudo-triangulation is a parti-tion of a planar region into pseudo-triangles. See [9]and [10] for surveys about theory and applications,with several interesting results that include combina-torial properties and counting of special classes, rigid-ity theoretical results, and representations as poly-topes, among others.Optimization problems related to pseudo-

triangulations are interesting under several optimalitycriteria. In this work, we consider the optimalitycriterion for pseudo-triangulations refereed as Mini-mum Weight. The weight of a pseudo-triangulationis the total length of their edges. Minimizing thetotal length is one of the main optimality criteriathat provides a quality measure. This formulation isknown as the Minimum Weight Pseudo-Triangulation(MWPT) problem. The complexity of the MWPTproblem is unknown an it is assumed to be inNP-hard class [6].Indeed, since no polynomial algorithm is known,

∗Email: [email protected] Research supported by Project Tec-nologías Avanzadas de Bases de Datos (22/F014), UniversidadNacional de San Luis, Argentina

†Email: [email protected] - Research supported by LIDIC,Universidad Nacional de San Luis, Argentina

‡Email: [email protected] - Research supported byESF EUROCORES programme EuroGICA. ComPoSe IP04-MINCINN Project EUI-EURC-2011-4306

approximate solutions of high quality are dicult toobtain by deterministic methods. Thus, we considerapproximation algorithms. These algorithms are ca-pable of obtaining approximate solutions to the op-timal ones and they can be easily implemented fornding good solutions in NP-hard optimization prob-lems [8].

In this work, we propose the use of Simulated

Annealing (SA) for nding high quality pseudo-triangulations of minimum weight. For this, we inves-tigate its application through an experimental studyand an extended statistical analysis of the results. Wehave implemented the algorithms involved and addi-tionally, generated our set of instances. Non para-metric statistical tests were applied for assessing theperformance of the algorithms implemented.

Our recent work on this research, [4] and [5], sum-marize successive stages of this research using a dier-ent metaheuristic. The preliminary results obtainedat the initial phase guided to apply a more method-ological approach in this research.

This paper is organized as follows. Section 1 de-scribes a general overview of the SA metaheuristic andthe main algorithms. Section 2 describes the experi-mental design, and Section 3 presents the experimen-tal evaluation and statistical analysis. Lastly, Section4 is devoted to the conclusions.

1 Simulated Annealing

Simulated Annealing applied to optimization prob-lems emerges from the work of S. Kirkpatrick et al.[7] and V. Cerný [2]. SA is based on the principles ofstatistical mechanics whereby the annealing processrequires heating and then slowly cooling a substanceto obtain a strong crystalline structure. This case isbased in an extension of local search (a trajectory-based approach) to solve combinatorial optimizationproblems. Without loss of generality, the strategy isgood for optimization problems, since SA has an ex-plicit strategy to escape from locally optimal solu-tions. SA introduces a control parameter, T , calledtemperature or cooling schedule, whose initial valueshould be high and should decrease during the searchprocess. The search process is done according to the

73

SA applied to the MWPT problem

execution of several iterations of the algorithm untila termination condition is achieved.In order to apply SA to the MWPT problem it is

necessary to dene some components of a SA by spec-ifying the following parameters: solution space S, ob-jective function f , neighborhood of a solution N (Si),initial solution S0, initial temperature T0, tempera-ture decrement rule R, number of moves at each tem-perature M(Tk) (length of the Markov chain), accep-tance function, and termination condition. The objec-tive of this study is assessing throughout a rigorousexperimental study the applicability and respectiveperformances of MWPT-SA and MWPT-SA-2P forthe MWPT problem. Next, we describe the commoncomponents taken into account in the experimenta-tion.

1.1 The MWPT-SA Algorithm

This algorithm is the result of two combined strate-gies: random walk and iterative improvement, com-monly named diversication and intensication. Thesearch has two phases. The rst phase consists of theexploration of the search space; however, this behav-ior is slowly decreased, leading the search to convergeto a local minimum, i.e., phase of iterative improve-ment. At each iteration, a neighbor of the neigh-borhood is randomly chosen. The neighborhood ofa pseudo-triangulation is obtained by application ofedge ips on two adjacent pseudo-triangles [1]. Themoves that improve the cost function are always ac-cepted. Otherwise, the neighbor is selected with agiven probability that depends on the current temper-ature. The basic outline is illustrated in Algorithm 1(MWPT-SA).

Algorithm 1 MWPT-SAGenerate an initial solution Si ∈ SSet the initial temperature to T0

k ← 0while termination condition not met do

c← 1while c < M(Tk) do

Choose Sj ∈ N (Si) ⊂ SEvaluate δ = f(Sj)− f(Si)if δ < 0 then

Si ← Sj

SaveBestSoFarSolutionelse

Si ← Sj with probability p(Tk, Si, Sj)end if

c← c+ 1end while

k ← k + 1Decrease temperature Tk

end while

Algorithm 1 (MWPT-SA) starts with an initial so-lution Si ∈ S, which can be randomly or heuristicallyconstructed. Then, it initializes the temperature withvalue T0. M(Tk) is the number of iterations for tem-

perature Tk. At each inner iteration, a new solutionSj ∈ N (Si) is randomly generated. If Sj is betterthan Si, then Sj is accepted as the current solution.Otherwise, the move from Si to Sj is an uphill move,and Sj is accepted with a probability computed ac-cording to the acceptance function. Finally, the valueof Tk is decreased at each outer iteration, controlledby variable k. The algorithm continues in this wayuntil the termination condition is met.

1.2 The MWPT-SA-2P Algorithm

This is an extend scheme, called MWPT-SA-2P al-gorithm, which was designed considering the coolingschedule of MWPT-SA. The cooling schedule is usedfor balancing between diversication and intensica-tion, by allowing to return to previous stages. In thismanner, the performance of MWPT-SA is improved.Indeed, escaping from the area of low quality in thephase of diversication was sometimes almost impos-sible for the basic algorithm. Then it is necessary tohave the possibility of exploring other regions of thesearch space. The basic outline is illustrated in Al-gorithm 2. The algorithm introduces an additionalvariable (named previous) in order to incorporate acontrol over the trajectory. This variable allows toknow which is the parameter setting of the algorithmrunning where the best so far solution has been found.

Algorithm 2 MWPT-SA-2PGenerate an initial solution Si ∈ SSet the initial temperature to T0

k ← 0while termination condition not met do

c← 1while c < M(Tk) do

Choose Sj ∈ N (Si) ⊂ SEvaluate δ = f(Sj)− f(Si)if δ < 0 then

Si ← Sj

SaveBestSoFarSolutionTpreviuos ← Tk

else

Si ← Sj with probability p(Tk, Si, Sj)end if

c← c+ 1end while

k ← k + 1if the best so far solution was updated then

Decrease temperature Tk

else

if it is the rst pass on Tk then

Return to previous temperature Tpreviuos andDo the second pass

end if

end if

end while

Algorithm 2 (MWPT-SA-2P) controls, before de-creasing the temperature, whether during the currenttemperature Tk the algorithm has found a better so-lution than the best so far solution. If not, the pro-cess returns to a state refereed as the previous tem-

74


perature, named Tpreviuos, where the best so far so-lution was found. From that state, the algorithmchooses moves for walking in other directions for ex-ploring other areas of the unexplored solution space.In this case, the SaveBestSoFarSolution process savesthe best solution found for all cycles so far and thecorresponding temperature Tpreviuos. The amount ofrepetitions to get through Tk was experimented, andwe conclude that a maximum of two is enough. As thetitle suggests, 2P in MWPT-SA-2P stands for doublepass, as the algorithm returns at most once on thepath traveled.

2 Experimental Design

Representation. The pseudo-triangulations are planarsubdivisions induced by planar embeddings of graphs.For their representation we use a Doubly-Connected

Edge List (DCEL) [3]. For evaluation purposes inSA, a solution must be transformed from the DCELinto a nxn matrix, where n is the number of points.

Objective Function. The weight of a pseudo-triangulation PT, named fw(PT ), is the sum of theeuclidean lengths of all the edges of PT.

Instances collection. An ad-hoc software was de-signed and implemented by the authors for generatingthe collection of problem instances, each one being aset P of n points in the plane. A collection of ten(10) instances of size n were generated, with n equalto 40/80/120. Each one is called LDni, 1 ≤ i ≤ 10.The points were randomly generated, uniformly dis-tributed, with coordinates x, y ∈ [0, 1000]. For imple-mentation purposes, there are non collinear points.

Parameter Settings for the SA algorithms. Theproposed algorithms were executed thirty (30) timesusing dierent random seeds for the complete col-lection of instances. The initial solution is apseudo-triangulation. We consider two types of ini-tial solutions for MWPT-SA-2P; the initial pseudo-triangulation is: a) a randomly generated solutionand b) a pseudo-triangulation obtained by the GPTalgorithm [4]. The initial temperature T0 dependson the number m of edges in the initial solution andthe objective function fw. T0 = m × l, where l isthe average length of the edges of the initial solution.The number of moves at each temperature N(Tk) isN(Tk) = Tk ensuring that the amount of moves isdirectly proportional to the actual temperature. Ineach case, for the Temperature decrement rule R,three dierent types of rules were considered: (i) FastSimulated Annealing (Tk+1 = T0

(1+k) ); (ii) Very Fast

Simulated Annealing (Tk+1 = T0

ek); and (iii) Geomet-

ric Decrease (Tk+1 = αTk, where α ∈ [0, 1]). TheGeometric cooling scheme had the best performanceaccording to previous experiments, therefore it waschosen for the study presented in this work. For this

cooling scheme, we consider α = 0.8, 0.9, and 0.95.The setting α = 0.95 was chosen due to its high per-formance. For the termination condition, the searchprocess is nished when the temperature is less thanor equal to Tf = 0.005.

Resources. The algorithms were implemented in C

and, for the statistical analysis, MATLAB was used.

3 Experimental Evaluation and

Statistical Analysis for the

proposed SA algorithms

This section shows the applicability of MWPT-SAand MWPT-SA-2P algorithms through experimentalevaluation. The initial solution can be a randomly so-lution generated, or a greedy pseudo-triangulation ob-tained by the GPT algorithm. SA using the last strat-egy for generating the initial solution can be consid-ered a hybrid approach. We reference the MWPT-SA-2P algorithm as MWPT-SA-2P-RPT (RPT stands forRandom Pseudo-Triangulation) or MWPT-SA-2P-GPT (GPT stands for Greedy Pseudo-Triangulation).The experimental and statistical study considers thementioned set of instances, the best objective, me-dian, average, and standard deviation values. Us-ing Kolmogorov-Smirnov test, we detect that the ob-tained values do not follow a normal distribution, thennon parametric statistical tests were applied to deter-mine if there is signicant dierence between algo-rithms. Wilcoxon rank-sum test was applied for al-lowing systematic pairwise comparisons and assessingwhether one of two samples of independent observa-tions came from populations with the same median.MWPT-SA and MWPT-SA-2P-RPT were compared,being the the null hypothesis rejected in all cases.The test rejected the null hypothesis of equal medi-ans with p-value less than 0.01 in all cases. Then,MWPT-SA-2P-RPT and MWPT-SA-2P-GPT werecompared, and also the the null hypothesis rejectedat all cases, showing the best performance of MWPT-SA-2P-GPT over MWPT-SA-2P-RPT. Figures 1 and2 display another perspective of the algorithms be-havior.

The comparison between MWPT-SA-2P-GPT andGPT algorithms can be observed in Figure 3.

It is also important to highlight that MWPT-SA-2P-GPT achieved objective values that exceed thevalues obtained by GPT by between 14% and 71%.The SA algorithms have best behavior with respect toGPT. The objective values of the solutions obtainedby the greedy algorithm have low quality with respectto those found by the SA algorithms. In summary,among the proposed algorithms, MWPT-SA-2P-GPTachieves the best performance, obtaining the highestquality solutions.

75

SA applied to the MWPT problem

Figure 1: Comparing SA algorithms w.r.t. best values

Figure 2: Comparing SA algorithms w.r.t. median values

Figure 3: Comparing MWPT-SA-2P-GPT and GPT

In addition, the computational eort of the pro-posed algorithms applied to the MWPT problem arecompared and analyzed. The metaheuristic algo-rithms consume much more computational resourcesthan a greedy algorithm, but their advantage is thatthey are capable of achieving solutions of much higherquality. Considering the results obtained and showedin previous subsections for all strategies, Table 1shows the average runtimes of the mentioned algo-rithms.

Table 1: Average runtimes of SA algorithms, adding thegreedy algorithm for the MWPT problem (in milliseconds).

Instance MWPT-SA MWPT-SA-2P-GPT GPT40 11679 27210 6980 25376 33239 83120 48342 51646 94

4 Conclusions

Our contributions show how SA can be applied tothe MWPT problem. We have developed MWPT-SA,MWPT-SA-2P-RPT, and MWPT-SA-2P-GPT algo-rithms. All claims were corroborated by the experi-mental study and the respective statical tests. Ourconclusions lead us to propose the use of MWPT-SA-2P-GPT for suitably solving MWPT.

References

[1] H. Brönnimann, L. Kettner, M. Pocchiola, andJ. Snoeyink. Counting and enumerating pointedpseudotriangulations with the greedy ip algorithm.SIAM J. Comput., 36:721739, 2006.

[2] V. Cerný. Thermodynamical approach to the travel-ing salesman problem: An ecient simulation algo-rithm. Journal of Optimization Theory and Applica-

tions, 1985.

[3] M. de Berg, M. V. Krevel, M. Overmars, andO. Schwarzkopf. Computational Geometry. Algo-

rithms and Applications. Springer-Verlag, 2000.

[4] M. G. Dorzán, E. O. Gagliardi, M. G. Leguizamón,and G. Hernández-Peñalver. Approximations on min-imum weight triangulations and minimum weightpseudo-triangulations using ant colony optimizationmetaheuristic. Fundamenta Informarticae, 119(1):127, 2012.

[5] E. O. Gagliardi, M. G. Dorzán, M. G. Leguizamón,and G. Hernández-Peñalver. Approximations on min-imum weight pseudo-triangulation problem using antcolony optimization. XXX International Conference

of the Chilean Computer Science Society. Jornadas

Chilenas de Computación, 2011.

[6] J. Gudmundsson and C. Levcopoulos. Minimumweight pseudo-triangulations. Comput. Geom.,38(3):139153, 2007.

[7] S. Kirkpatrick, C. Gelatt, and M. Vecchi. Optimiza-tion by simulated annealing. Science, 220:671680,1983.

[8] Z. Michalewicz and D. Fogel. How to Solve It: Mod-

ern Heuristics. Springer, 2004.

[9] M. Pocchiola and G. Vegter. Pseudo-triangulations:Theory and applications. In Symposium on Compu-

tational Geometry, pages 291300, 1996.

[10] G. Rote, F. Santos, and I. Streinu. Pseudo-

triangulations a survey. Contemporary Mathe-matics. American Mathematical Society, December2008.

76


A symbolic-numeric dynamic geometry environment for the

computation of equidistant curves

Miguel Á. Abánades∗1 and Francisco Botana†2

1CES Felipe II, Universidad Complutense de Madrid, 28300 Aranjuez, Spain2Dpto. Matemática Aplicada I, Universidad de Vigo, Campus A Xunqueira, 36005 Pontevedra, Spain

Abstract

A web-based system that determines point/curve andcurve/curve bisectors in a dynamic geometry systemin a completely automatic way is presented. The sys-tem consists of an interactive drawing canvas wherethe bisector is displayed together with the initialpoint/curve elements. Algebraic methods are used toprovide the equation of an algebraic variety contain-ing the bisector. A numeric approach is followed toprovide the graph of the semi-algebraic subset corre-sponding to the true bisector. It is based on the freedynamic geometry system GeoGebra and the opensource computer algebra system Sage.

Introduction

A Dynamic Geometry System (DGS) is a computerapplication that allows the on-screen drawing of (gen-erally) planar geometric diagrams and the manipula-tion of these diagrams by mouse dragging. The rststandard systems to appear (in the late 80's) wereCabri in France [10] and The Geometer's Sketchpad

in USA [9]. Nowadays special mention deserves Ge-

oGebra [6], whose free software model and eectivecommunity development has resulted in a spectacularworld wide distribution.From the beginning, DGS have been the paradigm

of new technologies applied to Math education. How-ever, most DGS rely on numeric computations and ap-proximate graphs, which make them prone to inaccu-racies. Moreover, their lack of symbolic tools preventDGS from realizing a thorough algebraic treatment ofgeometry.In this work we develop a symbolic treatment of

bisectors in the plane (locus set of points equidistantto two geometrical elements).In [1], symbolic algorithms to determine algebraic

∗Email: [email protected]. Research partially sup-ported by the project MTM2011-25816-C02-(01,02) funded bythe Spanish Ministerio de Economía y Competitividad.

†Email: [email protected]. Research partially supported bythe project MTM2011-25816-C02-(01,02) funded by the Span-ish Ministerio de Economía y Competitividad.

descriptions of point/curve and curve/curve bisectorsare described. In this note we present the implementa-tion of these algorithms in an open web-based systemin which symbolic capabilities are added to the DGSGeoGebra by connecting it (remotely) to the Com-puter Algebra System (CAS) Sage [13].The presented prototype is based on the remote

use of Sage rather than GeoGebraCAS [8], the CASavailable within GeoGebra, due to higher versatilityof Sage, that includes multiple specialized softwarepackages. The prototype wants to illustrate a gen-eral philosophy of DGS-CAS connection that we ndmore appropriate to implement with the most generalsystems possible. For this same reason, direct use ofPython from GeoGebra has not been considered.Together with these symbolic web applications, nu-

meric graphic alternative tools are provided to visual-ize a bisector when the exact symbolic computation ofits algebraic description is not computationally possi-ble.

1 Bisectors

Given two geometrical elements (points, curves, sur-faces, etc.), their bisector is the locus set of pointsequidistant to them. We consider bisectors of two ge-ometric objects O1 and O2 in the Euclidean 2-spaceE2, where each object is a point or an algebraic curve(see for instance Figure 1). Bisectors play an impor-tant role when constructing Voronoi diagrams, medialaxis transformations and in a variety of algorithms re-lated to shape decomposition (see [12]). A systematicstudy of plane bisectors can be traced back to [4],where the curves are parametrically described, and[7], where a set containing the bisector is obtained bysolving a system of nonlinear equations.Standard DG systems do not consider bisectors.

Besides the usual computation of the parabola viaits focus and directrix, there are not other primitivesfor such computations. Nevertheless, the bisector of apoint and a linear object in a DG environment can bepartially determined through an elementary locus op-eration. Since a bisector point is the center of a circle

77

Symbolic-numeric computation of equidistant curves

Figure 1: Bisector (dotted) of a line and an ellipse.

tangent to the linear object and passing through thepoint, the locus tool can suggest a graphical path con-taining the bisector. Similarly, following the symboliclocus approach in [2], an algebraic variety containingthe bisector can be found by solving the correspond-ing nonlinear polynomial system. We refer to this setas the algebraic or untrimmed bisector.In [1], a mixed algebraic-numeric approach for the

study of bisectors of points and lines within a DGSis sketched. More precisely, elimination techniquesare used for obtaining the detailed description of abisector. Since the complete bisector description fallsout of the algebraic setting, a numerical approach totrim the algebraic bisectors is shown to provide aneasy generation of bisectors.More concretely, the following is the algorithm pro-

posed to compute point/curve bisectors:

Input: curve c : f(x, y) = 0, point AStep 1: Compute (symbolically) the untrimmed(algebraic) bisector of A and cStep 2: Dene this object as an implicit curve dStep 3: Construct a point B on dStep 4: Construct the point D on c closest to BStep 5: If D and A are at the same distance fromB, then construct the point E, E = BReturn:

i) the locus graphic object truelocus of Ewhen B moves along dii) the equation(s) of the untrimmed bisector

Moreover, to compute the untrimmed bisector instep 1, a solution based on the remote use of a CASis indicated.The system presented here results mainly from the

implementation of these algorithms in a web systemwith a GeoGebra applet through an automatic con-nection with a remote Sage server.Determining the algebraic description of some bi-

sectors is computationally out of reach. To obtain thegraph of these bisectors we provide an alternative nu-merical method based on the dynamic color propertyof GeoGebra whose details can be found in Section2.2.

Figure 2: Algebraic bisector (red dotted) and truetrimmed bisector (black dotted) of point A(1, 0) andcurve y2 = x3.

2 System Description

The system consists of two main web applications cor-responding to the symbolic treatment of point/curveand curve/curve bisectors. Moreover, two auxiliaryweb applications showing a graphic illustration of abisector are also provided. They all have been in-cluded in a simple web page together with examplesand instructions freely available at [15].All four applications consist of a drawing canvas

where the bisectors are displayed together with theinitial elements. They all are based on the DGS Ge-oGebra and the CAS Sage.GeoGebra is a free DGS with multiple representa-

tions of objects in dierent windows: graphics, alge-bra, and spreadsheet. Its remarkable world wide usemakes GeoGebra a de facto standard in the eld. Sageis an open source CAS that integrates more than 100open-source packages (including Singular).

2.1 Symbolic web applications

After the user has input the point and the curve inthe applet, he/she just has to press the Find bisec-

tor button. The aleph.sagemath.org sagecell server[14] is then used to remotely obtain an algebraic va-riety containing the bisector whose graph is input inthe applet. To determine the true (trimmed) bisec-tor, a numeric comparison of distances is carried out.Figure 2 shows how the answer provides both the al-gebraic description of the bisector together with thetrue (trimmed) bisector.The algebraic treatment of the geometric data ob-

tained from the applet is done in Sage with some ad-hoc Python code composed of several hundred lines of

78


Figure 3: Bisector (white) of point (0, 2) and curvey4 + y2 − y − x10 − x3 − x = 0 (left) and bisector(white) of curves y2 −x3 = 0 and x3 + y4 = 9 (right).

code. More precisely, once the data are sent to Sage,the appropriate variables are initialized and the idealcorresponding to the task is generated. Singular (aCAS included in Sage with special emphasis on com-mutative algebra) is then called to basically computea Groebner basis for this ideal.It has to be noted that the integration of GeoGe-

bra and Sage has been implemented without loosinginteractivity. The DGS-CAS communication is syn-chronous. That is, changing an element in the Ge-oGebra construction does automatically trigger thecorresponding update on the Sage side.The structure of the symbolic web application for

the computation of simple curve/curve bisectors issimilar. It implements in a GeoGebra applet the an-alytic method sketched in [1] (algorithm 1, Section3).

2.2 Graphic web applications

In the case of bisectors, even point/curve bisectors insimple situations can be very involved. For instance,the bisector of the point (2, 2) and the curve y = x7+2is a polynomial of degree 20 with more than 150 terms.A curve/curve bisector is symbolically computed

as the intersection of two envelopes (see [3]), eachone previously obtained with an elimination proce-dure using Groebner bases. This makes the processtoo complex for the symbolic computation of bisectorsof curves other than lines and circles with current al-gorithms and standard hardware.When obtaining the algebraic description of a

point/curve or curve/curve bisector is not possible,two web applications providing a graphic illustrationof the bisector have been implemented, one for eachtype of bisectorThe idea of these graphic applications is to scan the

dierent 1-pixel points in the applet to change theirRGB color code. The color of a point P is changedaccording to a formula related to the distances to, forinstance, curves a and b, in such a way that the closerthe value distance(P, a) is to distance(P, b) the whiterthe point P becomes. Figure 3 shows a point/curveand a curve/curve bisector as graphed by the appli-cations.

This idea, based on the dynamic color property ofGeoGebra, was rst used by R. Losada [11]. Neverthe-less, this approach should be taken with care. Giventhe numerical character of the application, misleadinganswers can be returned.

3 Point/Curve Bisectors

In Figure 2 above we have already seen how the an-swer given by the application provides a complete de-scription of the bisector of a point and a curve, bothalgebraically and graphically. Here we give a roughdescription of the method.

If the curve is non-singular, obtaining the alge-braic bisector is a direct application of the elementarymethod for computing bisector points. Each bisectorpoint must lie on the intersection of the normal lineto the curve by a generic point on it, and the per-pendicular bisector of this point and the given one.Computing the locus of these points we get the alge-braic bisector, which will be trimmed in a subsequentstep. Note that if the initial point lies on the curveitself, the bisector is contained in the normal line tothe curve, as noted in [5]. However, if the curve issingular, the normal line will remain undened whenthe generic point is a singular one, thus including aspurious factor (the perpendicular bisector of the sin-gular point and the initial point) in the eliminationresult. Nevertheless, after the trimming process, allbut a nite number of points in this perpendicularbisector will be excluded.

If the initial point is a singular point on the curve,the normal line will be undened and the perpendic-ular bisector will be the whole plane, so the processreturns the ideal ⟨0⟩ after elimination. In this case,the trimming procedure does not have a proper vari-ety to trim, since the algebraic bisector is the wholeplane. Following [7], we exclude the singular pointsfrom our locus nding algorithm. In this way, the ap-plication returns some partial information about thebisector. For instance, we have a bi-dimensional bisec-tor for the curve y2 = x3 and the point (0, 0) as shownby the point/curve graphic application in [15] (Figure4, left). In this case, the symbolic web applicationprovides the algebraic description of the boundary ofthis bi-dimensional bisector (Figure 4, right).

As a conclusion, we note that the proposed appli-cations deal eciently with regular curves, while forsingular ones the results, although sometimes clever,must be taken with caution. The nal decision aboutbisectors in such cases should be guided by an ad-hocand specic study. The automatic determination ofthe bisectors in these cases is work in progress.

79

Symbolic-numeric computation of equidistant curves

Figure 4: 2D bisector of curve y2 = x3 and its cusp(0, 0) (left) and its boundary (right).

Figure 5: Algebraic bisector (dotted red) and truetrimmed bisector (dotted black) of circles (x + 4)2 +y2 = 8 and (x− 2)2 + y2 = 36.

4 Curve/Curve Bisectors

As mentioned above, for algebraic curves of low de-gree the symbolic application for the computation ofcurve/curve bisectors in [15] provides an algebraic va-riety where the bisector lies. This algebraic bisectoris then trimmed numerically within GeoGebra to dis-play the true bisector.Figure 5 shows the algebraic bisector and true bi-

sector of the intersecting circles (x+4)2+ y2 = 8 and(x− 2)2 + y2 = 36 as provided by the prototype.For curves of higher degree, the computation of the

algebraic bisector is (currently) computationally outof reach. In these cases we have already seen in 2.2how the graphic web application allows the displayof curve/curve bisectors for curves of any degree (seeFigure 3).

5 Conclusion

The prototype presented provides tools for thestudy of bisectors in the plane (point/curve andcurve/curve). Complete graphic information is pro-vided together with an exact algebraic descriptionwhen computationally possible.The system, web-based and interactive, shows the

power of the remote automatic connection of CAS

and DGS. Moreover, the exclusive use of free soft-ware, shows that it can be done without resorting toexpensive commercial systems.

References

[1] F. Botana. Computing bisectors in a dynamic geome-try environment. International Journal of Mathemat-

ical Education in Science and Technology, 44(2):299310, 2012.

[2] F. Botana and J. L. Valcarce. A software tool for theinvestigation of plane loci. Mathematics and Com-

puters in Simulation, 61(2):139152, 2003.

[3] R. Farouki and J. Johnstone. Computing point/curveand curve/curve bisectors. In R. B. Fisher, editor,Design and Application of Curves and Surfaces, num-ber V in The Mathematics of Surfaces, pages 327354. Oxford University Press, London, 1994.

[4] R. T. Farouki and J. K. Johnstone. The bisector of apoint and a plane parametric curve. Computer Aided

Geometric Design, 11:117151, 1994.

[5] R. T. Farouki and R. Ramamurthy. Degeneratepoint/curve and curve/curve bisectors arising in me-dial axis computations for planar domains withcurved boundaries. Computer Aided Geometric De-

sign, 15:615635, 1998.

[6] GeoGebra. http://www.geogebra.org, Last accessedMay 2013.

[7] C. M. Homann and P. J. Vermeer. Eliminating ex-traneous solutions in curve and surface operations.International Journal of Computational Geometry

and Applications, 1:4766, 1991.

[8] http://dev.geogebra.org/trac/wiki/GeoGebraCAS,Last accessed May 2013.

[9] N. Jackiw. The Geometer's Sketchpad v 4.0. KeyCurriculum Press, 2002.

[10] J. M. Laborde and F. Bellemain. Cabri Geometry II.Texas Instruments, Dallas, 1998.

[11] R. Losada. Propiedad de color dinámico en GeoGe-bra, http://geogebra.es/color_dinamico/color_

dinamico.html, Last accessed May 2013.

[12] M. Peternell. Geometric properties of bisector sur-faces. Graphical Models and Image Processing,62:202236, 2000.

[13] Sage. http://www.sagemath.org, Last accessed May2013.

[14] https://github.com/sagemath/sagecell, Last ac-cessed May 2013.

[15] http://webs.uvigo.es/fbotana/bisectors, Lastaccessed May 2013.

80



Oswin Aichholzer∗1, Thomas Hackl†1, Vera Sacristán‡2, Birgit Vogtenhuber∗1, and Reinhard Wallner1

1Institute for Software Technology, Graz University of Technology, Graz, Austria.2Departament de Matemàtica Aplicada II, Universitat Politècnica de Catalunya, Barcelona, Spain.

Abstract

We present a practical Java tool for simulating syn-chronized distributed algorithms on sets of 2- and 3-dimensional square/cubic lattice-based agents. ThisAgentSystem assumes that each agent is capable tochange position in the lattice and that neighboringagents can attach and detach from each other. Inaddition, it assumes that each module has some con-stant size memory and computation capability, andcan send/receive constant size messages to/from itsneighbors. The system allows the user to dene setsof agents and sets of rules and apply one to the other.The AgentSystem simulates the synchronized execu-tion of the set of rules by all the modules, and can keeptrack of all actions made by the modules at each step,supporting consistency warnings and error checking.Our intention is to provide a useful tool for the re-searchers from geometric distributed algorithms.

Introduction

Mainly due to their scalability, distributed algorithmsare a powerful tool for the control of self-organizingsystems. One of the most interesting examples of aeld of application is the control of modular roboticsystems and, in particular, the development of geo-metric algorithms for their locomotion, recongura-tion, and self-repair. When dealing with these sys-tems and algorithms, it is still often unaordable toactually implement and run the algorithms on a bigset of real prototypes and, in any case, it is recom-mended to simulate the behavior of the algorithmsprior to their actual physical implementation. Froma dierent viewpoint, frequently algorithmic resultsof theoretical nature are obtained but cannot imme-

∗Email: oaich,[email protected]. Research partiallysupported by ESF EUROCORES programme EuroGIGA -ComPoSe, Austrian Science Fund (FWF): I 648-N18.†Email: [email protected]. Research supported by the

Austrian Science Fund (FWF): P23629-N18 CombinatorialProblems on Geometric Graphs".‡Email: [email protected]. Research partially sup-

ported by projects MTM2012-30951, MTM2009-07242, Gen.Cat. DGR 2009SGR1040, and ESF EUROCORES programmeEuroGIGA, CRP ComPoSe: MICINN Project EUI-EURC-2011-4306, for Spain.

diately be translated into physical prototypes, as theymay require miniaturization or precision to a levelwhich is still out of reach. Having a simulator athand is then very convenient. In this paper we presentand describe the functionalities of a practical and verygeneral simulator that we hope will be useful in manydierent research contexts. Other multi-agents simu-lators are available with dierent scopes, as for exam-ple MASON [5]. Our tool supplies a framework forlattice-based modular robots. The instruction set forthe modules/agents is specic, simple, and intention-ally compact, as to make the use easy also for non-experts. In the following descriptions we present the2D information followed, if applicable, by additionalinformation needed for 3D in squared brackets.

1 The agents

The initial agents setting is stored in the leagents.txt. Each line of the le denes one agentby its initial (global) coordinates (mandatory) plus(optional) its state, its attachments and initializationsfor (some of) its counters. Optionally it is possible tostate the size of the universe in the agents le.

Universe size UminX,maxX,minY,maxY[,minZ,maxZ]To be positioned at the beginning of the le.

Initial (global) position x,y[,z]The initial position is written as integer x-, y- [andz-]coordinates, separated by a comma.

State S_____

The state of an agent consists of exactly 5 characters,written with a leading S.

Attachments A____[__]The attachments of an agent are written as A followedby 4 [6] booleans (0 for not attached, 1 for attached),in the order north, west, east, south[, above, below].

Counters C__ _____

Each agent has 25 [45] integer counters, C00,...,C24[,C25,...,C44], which can be set to any 16-bit in-teger between −32767 and 32767.

81


2 The rules

The denition of what a robot may do is stored inthe le rules.txt. Each rule denition consists of 4lines:1. the name of the rule,2. the priority of the rule,3. the precondition, and4. the postcondition.

The name is a nonempty string. Priorities are usedby each agent to decide which of the possibly severalrules that apply to its situation to execute. The prior-ity of a rule is a positive integer between 1 and 32767.Higher priorities win over lower ones. The precondi-tion denes whether or not an agent may apply therule. Finally, the postcondition denes the actions tobe performed when a rule is applied to an agent.

2.1 Precondition

The precondition of a rule is any boolean combinationof: compare priorities, check neighboring empty/lledpositions, check own connections, match states/textor counters/integers, and compare calculation resultswith counters, messages and integers.More precisely: a precondition is an AND combi-

nation of the following.

Neighbors N____[__]The situation of the direct neighboring positions(north, west, east, south[, above, below]). For eachof them, 0 denotes empty (no agent), 1 denotes lled(an agent), and * denotes indierent.

Empty position Edx,dy[,dz], EC__,dy[,dz],An empty position requirement. Written as an E,followed by the relative coordinates of the lattice po-sition required to be empty, separated by a comma.Alternatively instead of each value dx,dy or dz thename of any counter can be inserted, where a counterstarts with a C, followed by the two digits number ofthe counter.

Filled position Fdx,dy[,dz]A lled position requirement. The restrictions and thesyntax are the same as in the Empty position condi-tion.

Priorities P____[__]Compare the priority of (the applied rule/s) of thedirect neighboring agents (north, west, east, south[,above, below]) with the agents' own priority. For eachof them, < denotes that the priority of such agentneeds to be (strictly) smaller, = denotes smaller orequal, and * denotes indierent.

Smaller Priority Ldx,dy[,dz]A (strictly) less priority agent requirement. The L isfollowed by the relative coordinates of the agent re-quired to have smaller priority, separated by a comma.

The usage of counters is the same as in the Empty po-

sition condition.

Smaller or equal Priority Qdx,dy[,dz]A less or equal priority agent requirement. Syntaxand usage is analogous to the Smaller Priority condi-tion.

Attachments A____[__]The attachment states to the direct neighbors (north,west, east, south[, above, below]), where 0 denotesnot attached, 1 denotes attached, and * denotes in-dierent.

State S_____

The agent state can be required to match a simplepattern, where an asterisk matches any character.

State of a remote agent Tdx,dy[,dz],_____This is a combination of the Filled position and theState precondition. Written as a T, followed by therelative coordinates of the lattice position that needsto be lled, and ended by the state that the remoteagent must have. The usage of counters is the sameas in the Empty position condition.

(Text) messages from direct neighborsM*_____, MN_____, MW_____, ME_____, MS_____,[, MA_____, MB_____]Every agent has four [six] text messages from itsdirect neighbors (*=any, N=north, W=west, E=east,S=south[, A=above, B=below]), each consisting of ex-actly 5 characters. Any of these messages can be re-quired to match a pattern, where an asterisk matchesany character and at most four asterisks are allowed.

Numeric comparisons <(-)____ (-)____,>(-)____ (-)____, =(-)____ (-)____

In addition to its 25 [45] counters, every agenthas 4 ∗ 8 = 32 [6 ∗ 3 = 18] numeric messagesfrom its direct neighbors, denoted #N01,...,#N08,#W01,...,#W08, #E01,...,#E08, #S01,...,#S08 in2D, and limited to 3 counters per direction in 3D, in-cluding #A01,...,#A03, and #B01,...,#B03. Aster-isks can be used instead of a specic direction. Any ofthese numeric values can be required to fulll a com-parison with respect to any other such value or to anyfour digit number.

Remote numeric comparisonsVdx,dy,C___ (-)____, Wdx,dy,C___ (-)____

These options are only available in 2D. They allow tocompare the rst value with the second value. V indi-cates strictly smaller and W indicates smaller or equal.The rst numeric value is a counter from a remoteagent at relative coordinates dx,dy. It requires theagent to exist. The second numeric value can be acounter, a numeric message from a neighbor or anyfour digit number. See more details in the Numeric

comparisons description.

82


The following two operators enable generating anyboolean combination:

Parenthesis ( )

Group the expressions they surround.

Negation !

Negates the expression it precedes.

2.2 Postcondition

The postcondition of a rule denes the actions per-formed by an agent when it applies the rule. It isany AND combination of the following: change posi-tion, change attachments, modify state, compute andupdate counters, and send messages. More precisely:

Position change Pdx,dy[,dz]Move the agent to the given relative coordinates.Counters can be used as in the Empty position condi-tion.

Attachments A____[__]For each of the four [six] possible directions (in thealready described order), the possibilities are: 0 de-tach (if attached) before moving, and stay detachedafterwards; 1 detach (if attached) before moving, andattach afterwards (if possible); * detach (if attached)before moving, and attach afterwards if attached be-fore (and possible); + stay attached along the move-ment. In this case, attached agents are carried alongwith the moving agent.

State S_____

New state of the agent. An asterisk denotes that theaccording character remains unchanged.

(Text) messages to direct neighborsM*_____, MN_____, MW_____, ME_____, MS_____,[, MA_____, MB_____]Send messages to neighbors (* = all).

Calculations on counters and numerical mes-sages C___ _ ____ ____, #___ _ ____ ____

2D only:C___ _ dx,dy,C____ ____,#___ _ dx,dy,C____ ____,C___ _ C__,C__,C____ ____,#___ _ C__,C__,C____ ____

Every calculation action starts with a position towrite the result to (counter or outgoing message),followed by the operation to be performed, and two(readable) values on which the operation is per-formed. Possible operations are + (add), - (subtract),* (multiply), / (divide), M (modulo), A (maximum),and I (minimum). As values for an operation,either four-digit-numbers or internal counters (or oneexternal counter, only in 2D) or incoming numericalmessages can be used. The external counter isdened by rst indicating the coordinates (dx,dy orC__,C__) of the agent and then the counter.

Swap XN, XW, XE, XS, [XA, XB]Exchange the positions of two neighboring agents.Written as a X, followed by the desired swap neigh-bor.

3 The program ow

The program synchronously runs the rules on theagents. It starts by reading the initial setting as wellas the set of rules. At every step, the following oper-ations are performed in the order listed below. Alter-natively, the order of steps 2 and 3 can be transposedby the user, if desired. It is also possible for the userto make all rules not involving position changes to beapplied before those involving position changes.

1. Check and get valid rules. For all agents,check which rules would apply (ignoring priorities)and store valid rules sorted by priority. Store thehighest priority of valid rules as current priority andset the agents priority to open. For all open priorityagents sorted by priority, do until all agents have xedpriority: i) fetch current rules to current priority, andii) check priority-conditions for all rules. If they arefullled, set priority to xed. If a condition is not ful-lled, remove this rule from the specied agent and ifthe agents rule list is empty, reduce the current prior-ity to the highest priority of the remaining rules. If acircular dependency between rules on dierent agentsis detected, remove all related rules. Finally, for eachagent remove all rules with priority lower than thepriority of the agent.

2. Perform actions. For all agents, for all previ-ously stored rules for the agent, perform applicableactions in the following order: i) detach, ii) computeattachment decisions, iii) change position (includescollision detection test), iv) update attachments, v)update state, and vi) swap agents.

3. Compute calculations and send messages.For all agents and for all previously stored rules forthe agent, do all calculations (in the order they arelisted in the rule) and send numerical messages andall text messages to the post-oce. Then, deliver allmessages from the post-oce to their recipients.

4 The interface

The main window of the program consists of a menuand a tabbed panel with ve tabs, as can be seen inthe topmost portion of Figure 1.

Universe. This tab allows to visualize the agents asthey apply the rules. The algorithm can be visualized

83


step by step or can be let to run, it can be stopped,and it is also possible to jump one or more steps for-wards and backwards. In addition to the colors thatcan be used to distinguish the agents' states and theirattachments, clicking on an agent allows to show itsid, position, attachments, state, counter values, mes-sages, and current priority. Zooming and translatingthe scene is always possible. In the 3-dimensional sim-ulator rotations are also possible. Figure 1 shows ascreen shot of the universe of the 2D simulator, inwhich the information of one of the agents can beseen. The universe panel also shows the current num-ber of iterations, and all warning and error messages.

Figure 1: A screen shot of the program, showing thevisualization of a set of rules running on a set of agents.

Agents and Rules. The tab consists of two textpanels. The left one shows the agents le, the rightone shows the rules le. Both les can be indepen-dently loaded, modied and saved. Editing shortcutsare provided. When saving any of the les, inconsis-tencies and syntax errors are detected and marked.See Figure 2 for an illustration.

Figure 2: A screen shot of the panel showing the currentagents (left) and rules (right). An error detection is shown.

Log tabs. There are three log tabs, each showingthe corresponding le. The actions.log le stores theinformation of the rules applied by all agents at eachiteration. The positions.log le stores the completeinformation of all agents at each iteration (position,state, attachments, counters, etc.). The error.log lestores all error messages at each iteration.

Agents generator. This tab allows to graphicallygenerate or modify a set of agents, together with theirattachments, states and counters.

5 Implemented algorithms

We have designed and implemented a large set of dis-tributed algorithms, and we have run them on dif-ferent congurations of agents. The implemented al-gorithms cover tasks from self-organization to self-reconguration. Self-organization includes: choosinga leader, building a spanning tree, counting the num-ber of agents, and computing the minimum boundingbox. All these self-organization tasks refer to con-nected sets of agents. Details can be found in [4].Among the self-reconguration algorithms, we haveimplemented generic reconguration strategies for ar-bitrary connected shapes either assuming linear forceper module [4], inspired by the centralized algorithmproposed in [1], or only constant force [3], follow-ing [2]. In addition, we have also implemented pathnding algorithms, as well as some screen-saver-likeamusement ones.

6 Conclusion

Our simulator is robust, and it is our strong beliefthat it will be useful to researchers wishing to runexperiments on a wide range of distributed algorithmsfor self-organizing agents. For practical purposes thesystem scales linearly in nk, where n is the number ofagents and k is the number of rules.We therefore oer both the simulator and the

aforementioned examples to the scientic community.They can be downloaded from the web page [6], whichalso includes i) the source les, ii) a user guide, andiii) the details of the already implemented algorithms.

References

[1] G. Aloupis, S. Collette, M. Damian, E. D. Demaine,R. Flatland, S. Langerman, J. O'Rourke, S. Ra-maswami, V. Sacristán, S. Wuhrer, Linear recong-uration of cube-style modular robots, Computational

Geometry Theory and Applications, 42, 6-7 (2009),652663.

[2] F. Hurtado, E. Molina, S. Ramaswami, V. Sacristán,Distributed universal reconguration of 2D lattice-based modular robots, in: Proc. 29th European Work-

shop on Computational Geometry, 2013, 139142.

[3] O. Rodríguez, Simulació de l'actuació distribuïda de

robots modulars, Degree thesis, Universitat Politèc-nica de Catalunya, Spain, 2013.

[4] R. Wallner, A System of Autonomously Self-

Recongurable Agents, Degree thesis, Graz Universityof Technology, Austria, 2009.

[5] http://cs.gmu.edu/∼eclab/projects/mason/Last visited: May 15, 2013.

[6] http://www-ma2.upc.edu/vera/AgentSystems/

84



József Balogh∗1, Hernán González-Aguilar†2, and Gelasio Salazar‡3

1Department of Mathematics, University of Illinois at Urbana-Champaign. Urbana, IL, United States.2Facultad de Ciencias, Universidad Autónoma de San Luis Potosí. San Luis Potosí, SLP, México.3Instituto de Física, Universidad Autónoma de San Luis Potosí. San Luis Potosí, SLP, México.

Abstract

Given a set P of points in Rd, a convex hole (alterna-tively, empty convex polytope) of P is a convex poly-tope with vertices in P , containing no points of Pin its interior. Let R be a bounded convex region inRd. We show that if P is a set of n random pointschosen independently and uniformly over R, then theexpected number of vertices of the largest hole of Pis Θ(log n/(log logn)), regardless of the shape of R.This generalizes the analogous result proved for thecase d = 2 by Balogh, González-Aguilar, and Salazar.

Introduction

Given a set P of points in Rd, a convex hole (alterna-tively, empty convex polytope) of P is a convex poly-tope with vertices in P , containing no points of P inits interior.Recently, we showed that the expected size of the

largest convex hole in a random n-point set in theplane is Θ(log n/ log log n) [3]. One anonymous ref-eree of this paper asked if this could be generalizedto d > 2 dimensions. Joe O’Rourke asked the samequestion in MathOverflow, and Douglas Zare repliedthat the Ω(log n/ log logn) lower bound carries overeasily to the d-dimensional case [10]. At the end ofhis reply, Zare wrote: “I don’t know whether theirharder upper bound of the same form also extends tohigher dimensions, but I suspect that it does.”Our aim in this note is to show that, indeed, the

upper bound also holds for higher dimensions. Thus,our main result is:

Theorem 1 Let d ≥ 2 be an integer, and let R be abounded convex region in Rd. Let Rn be a set of npoints chosen independently and uniformly at randomfrom R, and let Hol(Rn) denote the random variable

∗Email: [email protected]. Research supported by NSFCAREER Grant DMS-0745185.†Email: [email protected]. Research supported by

PROMEP.‡Email: [email protected]. Research supported by

CONACYT Grant 106432.

that measures the number of vertices of the largestconvex hole in Rn. Then

E(Hol(Rn)) = Θ

(log n

log logn

).

Moreover, a.a.s.

Hol(Rn) = Θ

(log n

log logn

).

The proof, which is an immediate consequence ofTheorems 2 and 3 below, follows very closely the mainideas of the proof of [3, Theorem 3]. Indeed, the strat-egy and the main ideas are so close that it seems bestto follow as closely as possible the structure of [3]. Aswe shall see below, some of the results proved in [3]follow without any modification to arbitrary dimen-sions. The main adaptations needed are:

1. a generalization of the results in [3, Section 2] tod > 2 dimensions, to approximate convex sets inRd with lattice polytopes; and

2. an adaptation to d > 2 dimensions of the re-sults on the probability that a random n-pointset is in convex position, from the exact resultsof Valtr [11, 12] in R2 to the asymptotic resultsof Bárány [6] in Rd, for any d ≥ 2.

The workhorse for the proof of Theorem 1 for ar-bitrary regions R is the following statement, whichtakes care of the particular case in which R is a par-allelotope.

Theorem 2 Let R be a parallelotope in Rd. Let Rn

be a set of n points chosen independently and uni-formly at random from R, and let Hol(Rn) denotethe random variable that measures the number of ver-tices of the largest convex hole in Rn. Then

E(Hol(Rn)) = Θ

(log n

log logn

).

Moreover, a.a.s.

Hol(Rn) = Θ

(log n

log logn

).

85


The other essential fact is that the order of magni-tude of the expected number of vertices of the largestconvex hole is independent of the shape of R:

Theorem 3 There exist absolute constants b, b′ withthe following property. Let R and S be bounded convexregions in Rd. Let Rn (respectively, Sn) be a set of npoints chosen independently and uniformly at randomfrom R (respectively, S). Let Hol(Rn) (respectively,Hol(Sn)) denote the random variable that measuresthe number of vertices of the largest convex hole in Rn

(respectively, Sn). Then, for all sufficiently large n,

b ≤ E(Hol(Rn))

E(Hol(Sn))≤ b′.

Moreover, there exist absolute constants c, c′ such thata.a.s.

c ≤ Hol(Rn)

Hol(Sn)≤ c′.

We remark that Theorem 3 is in line with the fol-lowing result proved by Bárány and Füredi [5]: the ex-pected number of empty simplices in a set of n pointschosen uniformly and independently at random froma convex set A with non-empty interior in Rd is Θ(nd),regardless of the shape of A.

Remark (Proof of Theorem 1). Theorem 1 is animmediate consequence of Theorems 2 and 3.

The proof of Theorem 2 is in Section 1. As weexplain in Section 2, the proof of Theorem 3 is totallyanalogous to the proof of [3, Theorem 2].We make a few final remarks before we move on to

the proofs. For the rest of the paper we let Vol(U)denote the volume of a region U in Rd. We also notethat, throughout the paper, by log x we mean the nat-ural logarithm of x. Finally, since we only considersets of points chosen independently and uniformly atrandom from a region, for brevity we simply say thatsuch point sets are chosen at random from this region.

1 Proof of Theorem 2We start by noting that if Q,Q′ are two regions suchthat Q′ is obtained from Q by an affine transforma-tion, then Hol(Qn) = Hol(Q′n). Thus we may as-sume without loss of generality that R is the isotheticunit area square centered at the origin.We prove the lower and upper bounds separately.

More specifically, we prove that for all sufficientlylarge n:

Pr

(Hol(Rn) ≥ 1

2

log n

log logn

)≥ 1− n−2. (1)

Pr

(Hol(Rn) ≤ d(2 + 2d2)

log n

log log n

)≥ 1−n−1. (2)

We note that (1) and (2) imply immediately thea.a.s. part of Theorem 2. Now the Ω(log n/ log logn)part of the theorem follows from (1), since Hol(Rn)is a non-negative random variable, whereas theO(log n/ log logn) part follows from (2), sinceHol(Rn) is bounded by above by n.Thus we complete the proof by showing (1) and (2).

Proof of (1)

Let Rn be a set of n points chosen at random fromR. We prove that a.a.s. Rn has an empty convexpolytope of size at least logn

2 log logn .Consider the 2-dimensional projection π :

Rd → R2 defined by (x1, x2, x3, x4, . . . , xd) →(x1, x2, 0, 0, . . . , 0). Note that π(Rn) is a set of npoints chosen (independently and uniformly) at ran-dom from the unit square. Thus it follows from Eq. (1)in [3] that a.a.s. π(Rn) has a convex hole H of size atleast logn

2 log logn . Clearly, π−1(H) is an empty convexpolytope of Rn of size at least logn

2 log logn .

Proof of (2)

Let Rn be a set of n points chosen at random fromR. We remark that throughout the proof we alwaysimplicitly assume that n is sufficiently large.We shall use the following easy consequence of

Chernoff’s bound. This is derived immediately, forinstance, from Theorem A.1.11 in [1].

Lemma 4 Let Y1, . . . , Ym be mutually independentrandom variables with Pr(Yi = 1) = p and Pr(Yi =0) = 1− p, for i = 1, . . . ,m. Let Y := Y1 + . . .+ Ym.Then

Pr(Y ≥ (3/2)pm

)< e−pm/16.

Let S be the isothetic d-cube of volume 3d, also (asR) centered at the origin.We need the following result on approximating con-

vex sets by lattice parallelotopes.

Claim A. For each positive integer d > 0 there ex-ist integers f1(d) and f2(d) with the following prop-erty. Let H be a convex set in Rd. Then there ex-ists a lattice parallelotope Q1 such that H ⊆ Q1 andVol(Q1) ≤ (f1(d)+1)Vol(H). Moreover, if Vol(H) ≥2d−1 · 1000/n, then there is a lattice parallelotope Q0

such that Q0 ⊆ H and Vol(Q0) ≥ (f2(d)− 1)Vol(H).

Sketch of Proof. By a the theorem of M. Balla [2],for every convex compact set H ⊂ Rd there exists aparallelotope P such that P ⊂ H ⊂ dP = P wheredP is the image of P under a homothety with ratio d.This implies that d−d Vol(P ) ≤ Vol(H) ≤ dd Vol(P ).For each vertex vi, i = 0, . . . , 2d, of P , let us denote

by Qvithe parallelotope with side length 2/n with

86


facets parallel to the facets of P that has vi as one ofits vertices and P∩Qvi = vi. Observe that each Qvi

contains a d-ball of diameter 2/n and for that, thereis a lattice point v′i in the interior of each Qvi

. Let Q1

be the convex hull of the points v′1, . . . , v′2d . Note thatd(vi, v

′i) ≤ 2

n

√d for each i = 1, . . . , 2d, this implies

that %(P , Q1) ≤ 2n

√d where %(·, ·) is the Hausdorff

metric. Then

Vol(Q1) ≤ Vol(P ) +2

n

√d · Surf(P ) + Vol(B)

≤ dd Vol(H) +2

n

√d · Surf(P ) + Vol(B)

≤ (f1(d) + 1)Vol(H)

where B is the d-ball of diameter 2n

√d and Surf(·) is

the volume (d− 1)-dimensional.Now, for each vertex wi, i = 0, . . . , 2d, of P con-

sider the parallelotope Qwiwith side length 2/n with

facets parallel to the facets of P that has wi as oneof its vertices and Qwi ⊂ P . Because each Qwi con-tains a d-ball of diameter 2

n , then there exists a latticepoint in each Qwi

, let w′i this point. The existence ofthese points is guaranteed provided that Vol(H) ≥2d−1 · 1000/n Let Q0 be the convex hull of the pointsw′1, . . . , w

′2d . Note that d(wi, w

′i) ≤ 2

n

√d for each

i = 1, . . . , 2d, this implies that %(P,Q0) ≤ 2n

√d. Then

Vol(P )− 2n

√d ·Surf(Q0)−Vol(B) ≤ Vol(Q0) This im-

plies

Vol(Q0) ≥ Vol(P )− 2

n

√d · Surf(Q0)−Vol(B)

≥ Vol(P )− 2

n

√d · Surf(P )−Vol(B)

≥ d−d Vol(H)− 2

n

√d · Surf(H)−Vol(B)

≥ (f2(d)− 1)Vol(H).

For the rest of the proof, for simplicity we definef3(d) := f3(d), where f1(d) and f2(d) are as in ClaimA.Since there are (9n + 1)d < (10n)d lattice points,

out of which (n + 1)d are in R, it follows thatthere are fewer than

((10n)d2d

)< (10n)d2

d

lattice d-parallelotopes in total, and fewer than

(nd

2d

)< nd2

d

lattice d-parallelotopes all of whose vertices are in R.

Claim B. With probability at least 1 − n−10 everylattice parallelotope Q with Vol(Q) < 20f3(d) log n/nsatisfies that |Rn ∩Q| ≤ (3/2) · 20f3(d) logn.

Proof. We note that, since (f1(d) + 1)/f2(d) > 1,it follows that 20f3(d) > d2d + 10. Let Q be a lat-tice parallelotope with Vol(Q) < 20 f3(d) log n/n, andlet Z = Z(Q) ⊆ R be any lattice parallelotope con-taining Q, with Vol(Z) = 20 f3(d) logn/n. Let XQ

(respectively, XZ) denote the random variable that

measures the number of points of Rn in Q (respec-tively, Z). We apply Lemma 4 with p = Vol(Z) andm = n, to obtain Pr(XZ ≥ (3/2) · 20 f3(d) log n) <e−(3/2)20 f3(d)/24 logn = n−(5/4)f3(d). Since Q ⊆ Z,it follows that Pr(XQ ≥ (3/2) · 20 f3(d) log n) <n−(5/4)f3(d). As the number of choices for Q is atmost (10n)d2

d

, with probability at least (1−(10n)d2d ·

n−(5/4)f3(d)) > 1−n−10, no suchQ contains more than(3/2) · 20 f3(d) log n points of Rn.

A polytope is empty if its interior contains no pointsof Rn.

Claim C. With probability at least 1−n−10, there isno empty lattice parallelotope Q ⊆ R with Vol(Q) ≥20 (d2d + 10) log n/n.

Proof. The probability that a fixed lattice paral-lelotope Q ⊆ R with Vol(Q) ≥ d2d + 10 log n/n isempty is (1 − Vol(Q))n < n−d2

d+10. Since there arefewer than nd2

d

lattice parallelotopes in R, it fol-lows that the probability that at least one of the lat-tice parallelotope with area at least (d2d + 10) log n/n(and hence with area at least 20 (d2d + 10) log n/n) isempty is less than nd2

d · n−(d2d+10)≤n−10.

For the rest of the proof, we let H be a maximumsize convex hole of Rn.

Claim D. With probability at least 1− n−10 we haveVol(Q1)<f3(d) log n/n.

Proof. Suppose first that Vol(H) < 2d−11000/n.Then Vol(Q1) ≤ 2d−1 · 1000(f1(d) + 1)/n. Sincethis is obviously smaller than f3(d) log n/n, in thiscase we are done. Now suppose that Vol(H) ≥2d−1 · 1000/n, so that Q0 (from Claim A) exists.Moreover, Vol(Q1) ≤ (f1(d) + 1)Vol(H). SinceQ0 ⊆ H, and H is a hole of Rn, it follows thatQ0 is empty. Thus, by Claim C, with probabil-ity at least 1 − n−10 we have that Vol(Q0) <(d2d + 10) log n/n. Now since Vol(Q1) < (f1(d) +1)Vol(H) and Vol(Q0) ≥ f2(d) · Vol(H), it followsthat Vol(Q1) ≤ (f1(d) + 1)Vol(Q0)/f2(d). Thuswith probability at least 1 − n−10 we have thatVol(Q1) ≤

((f1(d) + 1)(d2d + 10)/f2(d)

)log n/n =

f3(d) logn/n.

Claim E. For each fixed integer d > 0, there exista universal positive constant c2 := c2(d) with the fol-lowing property. Let K be any convex polytope in Rd.Then the probability that r points chosen at randomfrom K are in convex position is at most (c2 n

2d−1 )−n.

Proof. This is an immediate consequence of [6] (seefor instance [4, Theorem 2.1]).

87


Claim F. With probability at least 1 − 2n−2 therandom point set Rn satisfies that no lattice paral-lelotope Q with Vol(Q) < 20 f3(d) logn/n containsd(2 + 2d2) logn/(log log n) points of Rn in convex po-sition.

Proof. Let Q be a lattice parallelotope withVol(Q) < 20 f3(d) logn/n. By Claim B, with prob-ability at least 1 − n−10 we have |Rn ∩ Q| ≤ (3/2) ·20 f3(d) logn. Thus it suffices to show that theprobability that there exists a lattice parallelotopeQ with with |Rn ∩ Q| ≤ (3/2) · 20 f3(d) logn andd(2 + 2d2) logn/(log logn) points of Rn in convex po-sition is at most n−2.Let c2 := c2(d) be as in Claim E. Thus the expected

number of r-tuples of Rn in Q in convex position isat most(|Rn ∩Q|

r

)(c2r

2d−1

)−r≤(

(3/2) · f3(d) logn

r

)(c2r

2d−1

)−r≤(e · (3/2) · F2 log n

r

)r (c2r

2d−1

)−r<(c3 log n · r−1−

2d−1

)r,

where c3 := 3ef3(d)/2c2.Since there are at most nd2

d

choices for Q, itfollows that the expected total number of such r-tuples with r = d(2 + 2d2) logn/ log logn is at mostnd2

d ·(c3 log n · r−1−

2d−1

)r< n−2 (this last inequal-

ity follows from an elemenary but long manipula-tion). This completes the proof, since it follows thatthe probability that such an r-tuple exists is at mostn2.

To finish the proof of (2), recall that H is a maxi-mum size empty convex polytope of Rn, and thatH ⊆Q1. It follows immediately from Claims D and F thatwith probability at least 1−n−1 the parallelotope Q1

does not contain a set of d(2 + 2d2) logn/(log logn)points of Rn in convex position. In particular, withprobability at least 1 − n−1 the size of H is at mostd(2 + 2d2) logn/(log log n).

2 Proof of Theorem 3The proof of Theorem 3 is totally analogous to theproof of [3, Theorem 2]. Indeed, in that proof, es-sentially all the arguments are independent of thedimension. The only adaptation that needs to bedone is that we need a version of [3, Corollary 6] ford > 2 dimensions. We recall that [3, Corollary 6]

claims that if H is a closed convex set in R2, thenthere exist rectangles U,K such that U ⊆ H ⊆ K,Vol(U) ≥ Vol(H)/8, and Vol(K) ≤ 2Vol(H).A d-dimensional analogue of this statement follows

from the following result in [9]: if H is a convex bodyin Rd, then H contains a parallelotope P such thatsome translate of dP contains K. Indeed, this impliesat once that if H is a closed convex set in Rd, thenthere exist parallelotopes U,K such that U ⊆ H ⊆ K,Vol(U) ≥ Vol(H)/d, and Vol(K) ≤ d ·Vol(H).

References[1] N. Alon and J. Spencer. The probabilistic method,

3rd. Edition. Wiley, 2008.

[2] M. Y. Balla, Approximation of convex bodies by par-allelotopes, Internal Report IC/87/310, InternationalCentre for Theoretical Physics, ICTP, Trieste, 1988

[3] J. Balogh, H. González-Aguilar, and G. Salazar,Large convex holes in random point sets. Comp.Geom. 46 (2013), 725–733.

[4] I. Bárány, Random points, convex bodies, lattices.Proceedings of the International Congress of Mathe-maticians, Vol. III (Beijing, 2002), 527–535. HigherEd. Press, Beijing, 2002.

[5] I. Bárány and Z. Füredi, Empty simplices in Eu-clidean space, Canad. Math. Bull. 30 (1987) 436–445.

[6] I. Bárány, A note on Sylvester’s four-point problem.Studia Sci. Math. Hungar. 38 (2001), 73–77.

[7] I. Bárány, Sylvester’s question: the probability that npoints are in convex position, Ann. Probab. 27 (1999),2020–2034.

[8] I. Bárány and P. Valtr, Planar point sets with a smallnumber of empty convex polygons. Studia Sci. Math.Hungar. 41 (2004), 243–266.

[9] G.D. Chakerian and S. K. Stein, Someintersection properties of convex bodies.Proc. Amer. Math. Soc. 18 (1967), 109–112.

[10] J. O’Rourke, Empty convex polytopes for randompoint sets.http://mathoverflow.net/questions/107875(2012).

[11] P. Valtr, Probability that n Random Points are inConvex Position. Discrete and Computational Geom-etry 13 (1995), 637–643.

[12] P. Valtr, The Probability that n Random Points in aTriangle Are in Convex Position. Combinatorica 16(1996), 567–573.

88



Ruy Fabila-Monroy∗1, Clemens Huemer†2, and Eulàlia Tramuns‡2

1Departamento de Matemáticas, Cinvestav-IPN2Departament de Matemàtica Aplicada IV, Universitat Politècnica de Catalunya, BarcelonaTech

Abstract

In 1979 Conway, Croft, Erd®s and Guy proved thatevery set S of n points in general position in the plane

determines at least n3

18 −O(n2) obtuse angles and also

showed the upper bound 2n3

27 − O(n2) on the mini-mum number of obtuse angles among all sets S. Weprove that every set S of n points in convex position

determines at least 2n3

27 − o(n3) obtuse angles, hencematching the upper bound (up to sub-cubic terms) inthis case. Also on the other side, for point sets withlow rectilinear crossing number, the lower bound onthe minimum number of obtuse angles is improved.

Introduction

A point set S in the plane is in general position ifno three points of the set lie on a common straightline. Throughout, all considered point sets S will bein general position in the plane and |S| = n. An angleabc at b determined by three points a, b, c of S is ob-tuse if it is greater than π

2 . Prominent problems andresults on obtuse and acute angles in point sets canbe found in [4]. Here we are interested in the numberof obtuse angles determined by point sets S. Conwayet al. [3] proved that the minimum number of obtuse

angles among all sets S is between n3

18 − O(n2) and2n3

27 − O(n2). In this note we prove that point sets

S in convex position determine at least 2n3

27 − o(n3)obtuse angles. Interestingly, this matches (up to sub-cubic terms) the upper bound example from [3]. We

conjecture that 2n3

27 is indeed the right order of magni-tude for the minimum number of obtuse angles. Pointsets in convex position are characterized as the pointsets that maximize the rectilinear crossing number.The rectilinear crossing number cr(S) of a point setS equals the number of convex quadrilaterals with

∗Email: [email protected]. Partially sup-

ported by Conacyt of Mexico, Grant 153984.†Email: [email protected]. Partially supported by

projects MEC MTM2012-30951 and Gen. Cat. DGR

2009SGR1040 and ESF EUROCORES programme EuroGIGA,

CRP ComPoSe: grant EUI-EURC-2011-4306, for Spain.‡Email: [email protected]. Partially supported by

MTM2011-28800-C02-01 from Spanish MEC.

i j

ia

b

c

Figure 1: Alternately skipping i and j points in thepolygonal path for class (i, j).

vertices in S. Hence, an upper bound of(n4

)on the

rectilinear crossing number is obvious. As for pointsets with low crossing number, the current best lowerbound is 277

729

(n4

)+Θ(n3) [2] and there are point sets S

that only have cr(S) = 0.380488(n4

)+ Θ(n3) [1]. We

show that point sets S whose crossing number is not

too large, at most 23

(n4

), have more than n3

18 obtuseangles.

Proofs

Theorem 1 Every set S of n points in convex and

general position in the plane determines at least 2n3

27 −o(n3) obtuse angles.

Proof. First we consider the case when n is a primenumber; the case when n is not a prime number will betreated at the end of the proof. We label the points ofS from 0 to n−1 in counter-clockwise order. For threepoints a, b, c ∈ S in counter-clockwise order, we saythat the angle abc at point b is of class (i, j) if the openhalfplane bounded by the line through points a andb, and not containing point c contains i points of S,and if the open halfplane bounded by the line throughpoints b and c, and not containing point a contains jpoints of S; see Figure 1. (Then ab is an i-edge and bcis a j-edge.) Hence each angle dened by S belongsto some class (i, j), where 0 ≤ i + j ≤ n − 3. For i, jxed, i 6= j, we consider the polygon P that starts atpoint 0, visits points of S in counter-clockwise order,alternately skipping i points and j points of S, untilit returns to point 0 the second time. Three steps of

89


such a polygonal path of P are shown in Figure 1.Note that the polygon P is self-intersecting and canvisit vertices more than once.

• Claim: Each angle of class (i, j) and each angleof class (j, i) is encountered exactly once in P .

We modify P to obtain a new polygon P ′ by pair-ing two consecutive steps of P which skip i pointsand j points respectively; that is, we now movefrom a point m to point m + i + j + 2 mod n.Since n is a prime number, each non-zero ele-ment of the additive group Zn is a generator ofthe group; in particular also i + j + 2. This im-plies that P ′ returns to the starting point 0 afterit visited each point of S\0 exactly once. Wenow retrieve the original polygon P by splittingthe paired steps into steps skipping alternately ipoints and j points. It follows that each point ofS is visited twice in P , and each angle of class(i, j) and each angle of class (j, i) is encounteredexactly once in P .

• Claim: The rotation number [5] of the polygonP is i + j + 2.

The rotation number measures how many timesthe polygon turns around. Note that the under-lying point set is in convex position and all stepsare done in counter-clockwise order. The polygonvisits each vertex m twice; from a point m thepolygonal path continues once to point m+ i+ 1mod n, and once to point m + j + 1 mod n; intotal the path advances i + j + 2 points fromm. Hence, summing over all n vertices, we count(i + j + 2)n steps between consecutive points ofthe point set in counter-clockwise order. n stepsbetween consecutive points describe one full turn.Thus the rotation number is i + j + 2.

• Claim: At least 2n− 3(i+ j + 2) angles of the 2nangles of classes (i, j) and (j, i) encountered in Pare obtuse.

For the sake of contradiction, suppose that Pcontains less than 2n−3(i+ j +2) obtuse angles.Then, P contains more than 3(i + j + 2) acuteor right angles. By an averaging argument, atleast one of the i+ j + 2 full turns of the polygoncontains more than three acute or right angles.But this is not possible, unless P contains fourright angles forming a 4-cycle contradicting nbeing a prime number.

Hence, each pair of classes (i, j) and (j, i) of an-gles, with i 6= j, contains at least 2n−3(i+j+2)obtuse angles. Summing over all possible valuesi, j we thus get the lower bound on the number

of obtuse angles in S

1

2

b 2n3 −2c∑i=0

b 2n3 −2−ic∑j=0,j 6=i

2n−3(i+j+2) =2n3

27−O(n2).

It remains to consider the case when n is not aprime number. In this case it suces to onlycount the number of obtuse angles in a subsetS′ of S consisting of np points, where np is thelargest prime number smaller than n. Since np >n− o(n), see e.g. [6], we get the lower bound onthe number of obtuse angles in S

2(n− o(n))3

27−O(n2) =

2n3

27− o(n3).

Lemma 2 Every set S of n points in general position

in the plane with rectilinear crossing number cr(S)

determines at least n3

12 −cr(S)n−3 −O(n2) obtuse angles.

Proof. We rst remark that the number of right an-gles formed by S is negligible for our purpose. In fact,it is enough to observe that no edge spanned by S isincident to more than two right angles, due to thegeneral position assumption. Hence we upper boundthe number of right angles by 2

(n2

). Each 4-tuple of

points in convex position forms at least one obtuseangle or four right angles; and each 4-tuple of pointsnot in convex position forms at least two obtuse an-gles. Thus, the total number of obtuse angles in S is

at least(cr(S)−2(n

2)/4)·1+((n4)−cr(S))·2

n−3 , where we divideby n − 3 because each obtuse angle is counted n − 3times. Simplifying gives the claimed bound.

References

[1] B.M. Ábrego, M. Cetina, S. Fernández-Merchant,J. Leaños, G. Salazar, 3-symmetric and 3-decomposable geometric drawings of Kn, Discrete

Applied Mathematics 158 (2010), 12401258.

[2] B.M. Ábrego, M. Cetina, S. Fernández-Merchant, J.Leaños, G. Salazar, On (≤ k)-edges, crossings, andhalving lines of geometric drawings of Kn, Discrete

and Computational Geometry 48 (2012), 192215.

[3] J.H. Conway, H.T. Croft, P. Erd®s, M.J.T. Guy, Onthe distribution of values of angles determined bycoplanar points, Journal of the London Mathemati-

cal Society (2) 19 (1979), 137143.

[4] P. Erd®s, Z. Füredi, The greatest angle among npoints in the d-dimensional Euclidean space, Annalsof Discrete Mathematics 17 (1983), 275283.

[5] B. Grünbaum, G.C. Shephard, Rotation and wind-ing numbers for planar polygons and curves, Trans-actions of the American Mathematical Society 322(1990), 169187.

[6] R. Guy. Unsolved problems in number theory, ThirdEdition, Springer, New York, 2004.

90


Stabbing simplices of point sets with k-flats

Javier Cano∗1, Ferran Hurtado†2, and Jorge Urrutia‡1

1Instituto de Matemáticas, Universidad Nacional Autónoma de México, Mexico City, Mexico.2Departament de Matemàtica Aplicada, Universitat Politécnica de Catalunya, Barcelona, Spain.

Abstract

Let S be a set of n points in Rd in general position.A set H of k-flats is called an mk-stabber of S if therelative interior of any m-simplex with vertices in Sis intersected by at least one element of H. In thispaper we give lower and upper bounds on the size ofminimum mk-stabbers of point sets in Rd. We studymainly mk-stabbers in the plane and in R3.

Introduction

A set x0, . . . , xk ∈ Rd of k+1 points is called linearlyindependent if there is no linear combination λ0x0 +· · · + λkxk = 0 in which at least one λi 6= 0, k ≤ d.A set of points x0, . . . , xk ∈ Rd is called affinelyindependent if the set x1−x0, . . . , xk−x0 is linearlyindependent, k ≥ 1. A set consisting of a single pointwill also be considered as affinely independent. Anaffine combination of a set of k+1 affinely independentpoints in Rd is a linear combination λ0x0 + · · ·+λkxk

of x0, . . . , xk such that λ0 + · · · + λk = 1. The setof affine combinations of k + 1 affinely independentpoints of Rd is called a k-flat. In particular, a (d−1)-flat of Rd will be called a hyperplane. Following ourintuition, a 1-flat is a line, and a 2-flat is a plane. Ak-flat is isomorphic to the k-dimensional space Rk. Apoint set S in Rd is in general position if any subset ofS with at most d+1 elements is affinely independent.An m-simplex of Rd is the convex hull of a set of

m + 1 affinely independent points in Rd, m ≤ d. Forexample, in R2 a 0-simplex is a point, a 1-simplex isa segment and a 2-simplex is a triangle.Given a k-flat h and an m-simplex P of Rd, we say

that h stabs P if h intersects the relative interior of P.A set of k-flats H is called an mk-stabber of a set ofpoints S if every m-simplex induced by the elements

∗Email: [email protected]. Research supported byproject number 178379 Conacyt, México.†Email: [email protected]. Research partially sup-

ported by projects MINECO277 MTM2012-30951, Gen. Cat.DGR2009SGR1040, and ESF EUROCORES programme Eu-roGIGA, CRP ComPoSe IP04: MICINN Project EUI-EURC-2011-4306.‡Email: [email protected]. Research supported by

project number 178379 Conacyt, México..

of S is stabbed by at least one element of H.For example, in the plane a set of points Q (respec-

tively lines) is a 30-stabber (respectively a 31-stabber)of point set S if any triangle with vertices in S con-tains an element of Q in its interior (respectively isintersected by a line in Q).In this paper we study the following problem: Given

two integers k < m, k < d, and m ≤ d+ 1, and a setS of n points in Rd in general position, how many k-flats are needed to stab all the m-simplices generatedby the elements of S?In this paper we focus mainly on stabbing all the

r-simplices of point sets on the plane or in R3, andgive some results for higher dimensions.The problem of finding sets of points that stab

all the triangles of a point set has been studied byKatchalsky and Meir [4], and independently by Czy-zowicz, Kranakis, and Urrutia [2]. They prove thatfor any point set S in the plane in general positionwith n elements, such that its convex hull has c ele-ments, the set of triangles of S can be stabbed withexactly 2n − c − 2 points, and that such a bound istight, as any triangulation of S has 2n−c−2 triangles.Stabbers for other convex holes, such as quadrilateralsand pentagons, have also been studied [1].Given a point set S in Rd, we define fm

k (S) as thesize of the smallest mk-stabber of S. We define fm

k (n)as the largest value that a mk-stabber can have overall the point sets S of Rd with n elements. With thisterminology, the preceding result in the plane trans-lates to f2

0 (S) = 2n− c− 2.In this paper we show upper and lower bounds for

fmk (n), for point sets on the plane and R3 that aretight up to a constant.

1 Well-separated sets and gener-alized ham-sandwich cuts

A family of convex sets C0, . . . , Ck of Rd, k ≤ d, iswell separated if for any choice of xi ∈ Ci, the set ofpoints x0, . . . , xk is affinely independent. A familyP0, . . . , Pk of finite point sets in Rd is well separatedif their convex hulls Conv(P0), . . . ,Conv(Pk) are wellseparated, k ≤ d.

91

Stabbing simplices with k-flats

Notice that in particular, this implies that for anypair of different indexes i, j ≤ k, the convex hulls ofPi and Pj do not intersect.Let P0, . . . , Pk be a family of pairwise disjoint finite

point sets in Rd, k ≤ d. Given positive integers ai ≤|Pi|, an (a0, . . . , ak)-cut is a hyperplane h for whichh ∩ Pi 6= ∅ and |h+ ∩ Pi| = ai, 0 ≤ i ≤ d, where h+ isthe half-space bounded below by h.The following result by Steiger and Zao will be use-

ful to us:

Theorem 1 ([8]) Let P0, . . . , Pd be well-separatedpoint sets in Rd such that P0 ∪ · · · ∪ Pd is in generalposition, and let a0, . . . , ad be positive integers suchthat ai ≤ |Pi|. Then there is a unique (a0, . . . , ad)-cutof P0, . . . , Pd.

Finally, we recall the almost folklore result aboutham-sandwich cuts:

Theorem 2 (Ham-sandwich theorem, [9])Every d finite sets in Rd can be simultaneouslybisected by a hyperplane. A hyperplane h bisects afinite set P if each of the open half-spaces defined byh contains at most b |P |2 c points of P .

Now we have all the tools that we will use to givean algorithm for constructing a worst-case optimalmk-stabber for point sets in the plane and in the 3-dimensional space.

2 Stabbers in the plane

We start by studying the following problem: Howmany lines are necessary to stab the set of line tri-angles (segments) determined by triples (pairs) of el-ements of a point set S in the plane? In our previousterminology, determine lower and upper bounds forf11 (n) and f2

1 (n) for point sets on the plane. We firstprove:

Theorem 3 For any set S of n points in the planer ≤ f2

1 (S) ≤ dn4 e, where r is the smallest number such

that n2 ≤ 2 + 2 + 3 + · · ·+ r. Our bounds are tight.

Proof. The lower bound follows from the fact that rlines, no three of which intersect at a point, divide theplane into 2 + 2 + 3 + · · ·+ r convex regions, and if aset of r lines stabs all of the triangles with vertices inS, then S has at most two elements in each of theseregions.We now prove that f2

1 (S) ≤ dn4 e. To this end, we

now show how to obtain a set H with dn4 e straight-

lines that is a 21-stabber of S. To prove our result, itis sufficient to find a set H with dn

4 e lines such thatany cell of the arrangement generated by H containsat most two elements of S.

First, we put in H one straight line ` that separatesS into two subsets of size dn

2 e and bn2 c respectively.

Refer to these sets as S1 and S2. Clearly, these sets arewell separated, and by Theorem 1 we can find a (2, 2)-cut for S1 and S2. This cut is a line `1 that leavestwo elements x1, x2 of S1 and two elements x′1, x′2 ofS2 above it. These points are separated from the re-maining points of S by ` and `1, if `1 contains someelement of S1 or S2 move it slightly. Thus any trian-gle containing at least one vertex in x1, x2, x

′1, x′2 is

intersected by ` or `1.We repeat this recursively on S1 \x1, x2 and S2 \x′1, x′2, dn

4 e − 2 times, obtaining a set of dn4 e lines

(including ` and `1) that is a 21-stabber of S.To show that our bound is tight, let S be a

set of n points in the plane in convex position la-beled p0, p1, . . . , pn−1 in counterclockwise order, withn even. Consider the set of n

2 triangles pipi+1pi+2, ieven, and i ≤ n

2 for every even value of i. Observethat any two of these triangles intersect at exactly onevertex, for n ≥ 6.We claim that any straight line stabs at most 2 of

these triangles. This is true since every such trianglehas exactly two edges of the convex hull of Pn; thenany straight line stabbing one such triangle stabs atleast one edge of the convex hull. Any straight linestabs at most two edges of the convex hull, so then itcan stab at most two of such triangles. Thus to staball of those triangles we need at least one straightline for every two triangles. Observe next that if theelements of S are in convex position, then to stab theedges of the convex hull of S we need at last dn

4 e lines.Thus our upper bound is tight.

The key property of H in the proof of this result isthe fact that any cell of the arrangement induced byH contains at most two elements of S. Observe thatif instead of finding a (2, 2)-cut for S1 and S2, we use(m− 1,m− 1)-cuts, then we obtain a set of lines suchthat any cell of the arrangement generated by themcontains at most m − 1 elements of S. An m islandof S is a subset S′ of S with exactly m elements andsuch that the convex hull of S′ contains no elementof S \ S′ in its convex hull. The next result followseasily from the proof of Theorem 3.

Theorem 4 For any set S of n points in the plane,d n

2(m−1)e lines are always sufficient and sometimesnecessary to stab all of the m islands of S.

In particular, for m = 2, this proves that f11 (S) ≤

dn2 e; that is, to stab the segments of S we need at mostdn

2 e lines. It is easy to see that the bound d n2(m−1)e

in Theorem 4 is achieved for point sets in convex po-sition.We close this section by observing that the problem

of calculating f11 (S) is equivalent to calculating the

minimum number of lines such that in each face of

92


the arrangement defined by these lines, there is atmost one element of S. This problem is known as theshattering problem and its decision version is knownto be NP -complete [5].

3 Stabbers in the space

We consider now the problem of stabbing the tetra-hedra induced by an n point set S in R3 using sets ofplanes. Our approach is similar to that of the previoussection.For the lower bound we will use the following set

of n points in the plane. Let f : R → R3, f(a) =(a, a2, a3). f(R) is known as the momentum curve inR3. Clearly, any plane in R3 intersects f(R) in at most3 points. Let Cn = p0, . . . , pn−1 with pi = f(i).Since any plane intersects f(R) in at most 3 points,Cn is in general position. Given any segment pipj ,let f([i, j]) be the curve containing the points f(x),with i ≤ x ≤ j. We call f([i, j]) the shadow of thesegment pipj in the momentum curve. The followingobservation will be useful.

Observation 1 Let pi, pj ∈ Cn. Then any plane in-tersecting the relative interior of the segment pipj in-tersects its shadow.

Consider the set t0, . . . , tbn−13 c of tetrahedra, such

that the vertex set of ti is p3i, p3i+1, p3i+2, p3i+3.Observe that the interiors of these tetrahedra are pair-wise disjoint, and that ti and ti+1 share exactly onevertex, namely p3i+3.

Observation 2 If a plane intersects the interior of atetrahedron, then it intersects the relative interior ofone of its edges.

We define the shadow of ti as the shadow of the seg-ment p3ip3i+3 which is in fact the union of the shadowsof the edges of ti. Then the relative interiors of theshadows of t0, . . . , tbn−1

3 c are pairwise disjoint. The

next lemma follows now directly from Observations 1and 2:

Lemma 5 If a plane stabs ti, then it intersects therelative interior of its shadow.

Since any plane intersects the momentum curve atmost three times, it follows that any plane intersectsat most three elements of t0, . . . , tbn−1

3 c.

By Observation 1, any plane stabbing ti must in-tersect its shadow; thus any any plane stabs at mostthree elements of t0, . . . , tbn−1

3 c.

Thus we have:

Lemma 6 f32 (n) ≥ dn−1

9 e.

We will now prove that f32 (n) ≤ dn

9 e + 2. To thisend, we are going to give an algorithm that splits anypoint set S into a a family of well-separated familiesof subsets of S.First find two parallel planes π1 and π2 such that

they split S into three subsets S1, S2 and S3 such that||Si| − |Sj || ≤ 1, i 6= j ≤ 3. That is, each Si containsbn

3 c or dn3 e elements.

We now apply the ham-sandwich theorem in R3

to find a plane π3 that simultaneously bisects S1, S2

and S3. Let S+i be the subset of Si that lies above

π3 and S−i the one that lies below π3, with i = 1, 2, 3.This gives us two disjoint well-separated families F =S+

1 , S−2 , S

+3 and F ′ = S−1 , S

+2 , S

−3 . Similarly to

the algorithm in the previous section, iteratively find(3, 3, 3)-cuts for F , removing the nine points abovethe cut until each set in F contains at most threepoints, and adding the cut planes to the 32-stabber.Then repeat the same process for F ′.It is now easy to see that the set of planes we obtain

has at most dn9 e + 2 elements, and that they form a

stabber of all the tetrahedra with vertices in S. Thuswe have proved:

Theorem 7 dn−19 e ≤ f

32 (n) ≤ dn

9 e+ 2.

Observe that using similar arguments we can provethat f2

2 (n) ≈ n/6 (stabbing triangles with planes) andf12 (n) ≈ n/3 (stabbing segments with planes).

3.1 Stabbing 2-simplices of point setswith lines

For the problem of stabbing triangles with lines, it iseasy to see that in some instances, we need at least2n−4

2 lines. Indeed, consider any set of points S inR3 with n points in convex position. Observe thatthe convex hull of S contains exactly 2n−4 triangles,and that any line intersects the interior of at most twoof these triangles. Thus we have proved that in R3,f21 (n) ≥ n− 2.We now prove that 2n−5 lines are always sufficient

to stab all of the triangles of S. To prove this, firstproject the elements of S into the real plane, thusobtaining a point set S′ in R2. It is known that if theconvex hull of S′ has c elements, it is always possibleto find a set of points Q with at most 2n − c − 2points that stabs all of the triangles of S′; see [2, 4].Each point q ∈ Q generates a vertical line `q passingthrough q. Since each triangle T of S projects to atriangle T ′ of S′, there is a line `q, q ∈ Q that stabsT ′ and thus T . Since in the worst case, the convexhull of S′ has three points, we have proved:

Theorem 8 In R3, f21 (n) ≤ 2n− 5.

93

Stabbing simplices with k-flats

3.2 Stabbing the 3-simplices of a pointset in R3 with points

We now turn our attention to the problem of stabbingthe set of tetrahedra generated by a point set S inR3 using sets of points. We will assume that no twoelements of S lie on a line vertical to the real plane.Consider again the momentum curve f(a) =

(a, a2, a3), and the set of points Cn = p0, . . . , pnas defined above. Let 0 ≤ i, j ≤ n such thati + 1 < j. Let Ti,j be the tetrahedra with vertex setpi, pi+1, pj , pj+1. It is well known [3] that the set oftetrahedra Ti,j , with i and j as above, have disjointinteriors, and form a tetrahedralization of the convexhull of Cn = p0, . . . , pn. Since any set of points thatstabs the tetrahedra of S contains at least one pointin each of these Ti,j , it follows that f3

0 (n + 1) is atleast

(n−1

2

).

We now prove an upper bound on f30 (n + 1). We

proceed as follows: For every two elements xi, xj ∈ S,place a point pi,j slightly above the midpoint of theline segment joining xi to xj . For every xi ∈ S, placea point pi slightly below xi.We now prove that the set of points containing all

the p′i,js and the pi’s stabs all the tetrahedra of S.Consider any four points xi, xj , xk, x` of S. Let H

be the convex hull of xi, xj , xk, x`. Two cases arise.In the first case, H projects to a convex quadrilateralin the plane. In this case, one edge of the convex hullof xi, xj , xk, x` is not visible from below. Supposew.l.o.g. that it is the edge joining xk to x`. Then thepoint pi,j stabs H.Suppose then that H projects to a triangle T on

the plane. Then one vertex of H, say xi, belongs tothe interior of T . Two sub-cases arise. In the firstsub-case, when we see H from below, xi is not visi-ble. In this case, the point pi stabs the convex hull ofxi, xj , xk, x`. In the remaining sub-case, pi,j stabsH.Thus we have that we can always choose

(n+1

2

)+ n

points that stab the tetrahedra of S. It is not hardto see that we can reduce the above number by 9 byobserving that if the line segment joining xi to xj isnot visible from below, and this line segment belongsto the convex hull of S; then the point pi,j is redun-dant. Also if a point xi belongs to the convex hull ofS and is visible from below, then pi is redundant.Thus we have proved:

Theorem 9(n−1

2

)≤ f3

0 (n+ 1) ≤(n+1

2

)+ n− 9.

4 Some thoughts on higher di-mensions

Some of the results in this paper; e.g. Lemma 5and Lemma 6, generalise easily to higher dimensions,

yielding the following result:

Lemma 10 In Rd, dn−1d2 e ≤ fd

d−1(n).

And in general:

Lemma 11 In Rd, dn−1dm e ≤ f

md−1(n).

To finish this paper, we conjecture:

Conjecture 1 fdm(n) ≈ n

m·d .

References[1] J. Cano, A. García, F. Hurtado, T. Sakai, J. Tejel,

J. Urrutia, Blocking the k-holes of point sets in theplane, submitted.

[2] J. Czyzowicz, E. Kranakis, J. Urrutia, Guarding theconvex subsets of a point set, in: 12th CanadianConference on Computational Geometry, Frederic-ton, New Brunswick, Canada, 2000, 47–50.

[3] B. Grünbaum, Convex Polytopes, Interscience Mono-graphs in Pure and Applied Mathematics, Vol. XVI,John Wiley & Sons, London-New York-Sydney.

[4] M. Katchalski, A. Meir, On empty triangles deter-mined by points in the plane, Acta Math. Hungar 51(1988), 323–328.

[5] R. Freimer, J. S. B. Mitchell, C. D. Piatko, On theComplexity of Shattering Using Arrangements, in:Proceedings of the 2nd Canadian Conference on Com-putational Geometry, 1990, 218–222.

[6] A. Pór, D. R. Wood, On visibility and blockers, J.Computational Geometry 1(1) (2010), 29–40.

[7] T. Sakai, J. Urrutia, Covering the convex quadri-laterals of point sets, Graphs and Combinatorics 38(2007), 343–358.

[8] W. Steiger, J. Zhao, Generalized Ham-SandwichCuts, Discrete and Computational Geometry 44(3)(2010), 535–545.

[9] H. Steinhaus, A note on the ham sandwich theorem,Mathesis Polska 9 (1938), 26–28.

94



Lluís Enrique∗1 and Rafel Jaume†2

1ETH Zürich, Switzerland2Freie Universität Berlin, Germany

Abstract

We introduce a class of solids that can be constructedgluing stackable pieces, which has been proven to haveadvantages in architectural construction. We derivea necessary condition for a solid to belong to thisclass. This helps to specify a simple sufficient con-dition for the existence of a stackable tessellation of agiven solid. Finally, we show the compatibility of ourmethod with some discretization techniques appear-ing in the literature.

Introduction

During the last decades, the increasing demand of ar-chitectural freeform shapes has motivated the studyof surface tesselations in tiles which can be manu-factured with a low economic impact. Substantialresearch has been done on planar tesselations withtriangular meshes, quadrilateral meshes [1, 2] and itsspecial cases of spherical and conical meshes [3]. Nev-ertheless, tessellating methods in non-standard non-planar panels have been rarely considered, since formost construction materials the production of suchbuilding components is very expensive. The inno-vative CASTonCAST project [4] proposes a systemwhich consists of two parts: a geometric method forthe construction of double curved shapes by means ofstackable tiles and an efficient fabrication technique ofcurved precast panels which relies on shaping buildingcomponents in stacks by using the previous compo-nent as a mold for the next one. The latter relies onusing the top surface of each component as part of themold for the next one, after applying a chemical bar-rier. Once the components have been fabricated instacks, they are detached and assembled by glueingthem through their lateral faces, forming a new solidcalled strip. This procedure features obvious advan-tages concerning storage and transport as well.We analyse the kind of solids that can be con-

structed using stacks under some reasonable assump-

∗Email: [email protected]. Research supported by ObraSocial "la Caixa".†Email: [email protected]. Research supported by

Obra Social "la Caixa" and the DAAD.

tions. More precisely, we focus on constructions wherethe tiles form a face-to-face tessellation whose adja-cency graph is a grid both in stack and strip forms, asin Figure 1. We also require the junction faces in thestrip configuration to be planar. After stating neces-sary conditions for a solid to admit a stackable tes-sellation as desired, we provide sufficient conditionsand a procedure to obtain the stacks that generateit. It turns out that the solids for which a stack-able tessellation exists are naturally defined, skippingsome technical details, as the ones generated sweep-ing a surface around an axis or in a fixed direction. Inaddition, we prove that, under certain conditions, aone-parametric family of stackable tessellations exists.Finally, we approach the problem of approximating atarget solid using polyhedral tiles.

Figure 1: Stacks (right) and associated strips (left)

We normally use capital letters for points, lower-case letters for vectors, lines and rays, and Greek let-ters for curves. We add a ∗ to the names in order toindicate the correspondences between the 2- and 3-dimensional elements of the constructions in Sections1 and 2. We will often omit the indices ranges, forease of reading. The curves and surfaces are consid-ered to have no self-intersections. Abusing notation,we denote the image set of a curve or a surface by itsname. Curves are assumed to be parametrized usingthe unit interval. We define the wedge spanned bytwo rays to be the cone defined by their supportinglines and containing an unbounded part of both rays.

95


1 Planar stacksBefore starting the analysis of the 3-dimensional case,we study a 2-dimensional object we call planar stack.Its construction is described in the next subsectionand illustrated by Figure 2.

1.1 ConstructionLet p and q be two infinite rays emanating from apoint K and spanning a wedge W of angle α ∈ (0, π)in R2. Let σ0, ..., σn be pairwise disjoint, simple curvescontained inW . Assume the indices correspond to theinverse order a ray shot from K would intersect them.Assume further that σi ∩ p = σi(0) and σi ∩ q = σi(1)and require that

d(σi(0), σi+1(0)) = d(σi−1(1), σi(1)), (1)

where d denotes the Euclidean distance in the plane.Let Ti denote the closed subset of W bounded by σiand σi+1. Each of these simply connected sets will becalled a tile and each σi will be called a separator. Thewhole construction will be referred to as a planar stackof angle α. It may also be required that the curvesσi are separable, i.e., that a tile can be moved awayfrom the next one by a rigid motion preventing themto overlap. A simple condition ensuring this propertyis that each σi is monotone in some direction but wewill ignore this requirement in this article.

p

q

K

W

α

σ0

σ1

σ2

σ3

σ4

c0(T0) c1(T1) c2(T2)c3(T3)γ

µ = RCα (γ)

T0

T1T2

T3

c1(T1)

Figure 2: A planar stack and the corresponding strip.

Note that condition (1) ensures that the tiles can berearranged gluing σi+1(0) with σi(1) and σi(0) withσi−1(1). This configuration will be called the stripassociated to the planar stack.It will be handy to define si : R2 → R2 to be the

congruence that maps Ti to its position in the strip,assuming s0 = Id. Given a point C ∈ R2 and an angleα, we define RCα to be the counterclockwise rotationof angle α and center C.

Theorem 1 If the strip⋃n−1i=0 si(Ti) associated to a

planar stack of angle α is a simply connected set, thenits boundary contains the curves γ =

⋃n−1i=0 si(σi+1)

and RCα (γ), for some point C ∈ R2.

Proof. The assumptions that all the σi start in pand end in q, are pairwise disjoint and have no self-intersections ensure that the tiles are topologicallydisks. Provided that

⋃n−1i=0 si(Ti) is simply connected,

the tiles must intersect only in the glued segments.In addition, condition (1) ensures that si(σi(1)) =si+1(σi+1(0)) and si(σi+1(1)) = si+1(σi+2(0)), fori ∈ 0, ..., n− 2. Consequently, it is clear that γ andµ =

⋃ni=1 si(σi) are contained in the boundary of the

strip. Observe now that when Ti+1 is glued next toTi, the two copies of σi+1 involved are congruent viaRCiα , for some point Ci. But RCi

α maps si(σi+1(1))

to si+1(σi+1(1)) and RCi+1α maps si+1(σi+2(0)) to

si+2(σi+2(0)), which are the same pair of points.Since for any pair of points P 6= Q and a given αthere is only one point O such that ROα (P ) = Q, ithas to be Ci = C for all i ∈ 0, ..., n − 1. Thus, wehave that µ = RCα (γ).

1.2 TessellationConstructing a planar stack and developing it can beused as a modelling tool. However, we are now inter-ested in the other direction of the procedure. That is,given a curve γ, an angle α and a center of rotation C,construct a planar stack such that, when reconfiguredinto strip, contains γ and RCα (γ) in its boundary. Sucha planar stack may not exist. If it does, it may not beunique. Indeed, there is in general a one-parametricfamily of possibly suitable planar stacks.Theorem 1 indicates that we need to construct a

stack of angle α if we aim to obtain a shape with γand RCα (γ) forming part of its boundary. Considerthe locus of points that see a counterclockwise angleα between γ(0) and RCα (γ(0)), i.e., the set

κ = P ∈ R2 : RPα (−−−→Pγ(0)) =

−−−−−−−→PRPα (γ(0)), (2)

where−−→XY is the ray starting at X and going through

Y . It is well known that κ is a circumference arcgoing from γ(0) to RCα (γ(0)). It is not hard to seethat C ∈ κ. Obviously, any candidate K to be thevertex of the stack must belong to κ and there is apossibly valid planar stack leading to our target foreach of such points.Consider a fixed K ∈ κ and let r0 be the ray con-

tainingK, passing through γ(0) and such that RCα (r0)touches but does not cross r0. We define ri = RCiα(r0)for i ∈ N. Let Wi be the wedge of angle α spanned byri and ri+1 and define also Ui to be the region in Wi

bounded by γ and RCα (γ). For simplicity, we assumethat γ does not intersect RC−α(r0). This last restric-tion is not necessary, but simplifies the technicalities.

96


C

Kκ

r0r1 r2

W1

U1

Wi

Uiri

Figure 3: Tessellation procedure.

Theorem 2 If r0, ..., rn intersect γ in a single point,the tiles U1, ...,Un−1 can be arranged as a planar stack.

Proof. We will prove that RC−iα(Ui)n−1i=1 is a setof cells tessellating a simply connected subset of W0.Observe first that Wi+1 = RCα (Wi), since rj+1 =RCα (rj) for all j ∈ N. For the same reason, r1, ..., rn+1

intersect RCα (γ) in a single point. Since γ and RCα (γ)do not intersect, the boundary of Ui is formed by asubcurve of γ and a subcurve of RCα (γ) which aremutually disjoint, one segment contained in ri andone contained in ri+1. It remains to be proven thateach tile matches the next one and they all fit in W0.This can be derived from the facts that RCα (Ui ∩ γ) =Ui+1 ∩RCα (γ) and that RC−iα(Wi) =W0.

Using the previous theorem, it is not hard to seethat under an additional technical condition, one canconstruct a stack containing the whole γ and RCα (γ) inits boundary. We state this in the following corollary,omitting its simple proof.

Corollary 3 If r0, ..., rl intersect γ in a single pointand rl+1 does not intersect γ, then there exists a pla-nar stack containing γ and RCα (γ) in its boundary.

1.3 Refinement

For obvious reasons, it can be useful to refine a stack-able tessellation. This feature will be essential in thediscussion concerning approximation in Section 3.Adopting the notation used in the previous section,

consider the angularly ordered rays r0 = r00, ..., rm0 =

r1 emanating from K and contained in W0. De-fine also rji to be RCiα(r

j0) for i ∈ 1, ..., n and

j ∈ 0, ...,m. Let W ji be the wedge spanned by

r0

r1

r10

r20

W 10 W 1

1

r11 r21r2

Figure 4: Shaded tiles can be arranged in stack form.

rji and rj+1i . Define also U ji to be the space in W j

i

bounded by γ and RCα (γ).

Proposition 4 If r00, ..., rm0 = r01, r11, ..., r

ml intersect

γ in a single point, the tiles U j1 , ...,U jn−1 can be ar-ranged as a planar stack, for j ∈ 0, ...,m− 1.

Proof. The way we defined rji ensures thatRC−iα(W

ji ) = W j

0 . Therefore, the arguments in theproof of Theorem 2 apply to each j individually.

Note that, in addition, the m stacks obtained fromthe refined tessellation can be obtained by cutting theoriginal stack by straight radial cuts.The results presented in this section can easily be

extended to the case where K is thought of as a pointat infinity. That is, we consider the wedge W to bethe space between two parallel lines. Then, the twocopies of γ appearing in the boundary of the strip arerelated by a simple translation. Conversely, given acurve and a translated copy of it, a one-parametricfamily of possible stacks can be again evaluated. Inthis case, we may consider κ to be the set of directionsof the plane, except for the direction of W . We canthen cut the curves by a set of equally spaced linesin any direction u ∈ γ. This case is specially inter-esting because it generates offset shapes. Moreover,requiring that γ is monotone in u is sufficient to en-sure the existence of the corresponding stack with theadditional separability property.

97


2 Solid stacks

We consider now a 3-dimensional extension of the pre-vious results, which is the actual target of this work.In this way, we model a meaningful subset of thesolids constructible using the CASTonCAST fabrica-tion technique.The initial object is now the space W ∗ between

two non-parallel planes p∗ and q∗, which we furtherbound by two planes orthogonal to K∗ = p∗ ∩ q∗.Cutting W ∗ with pairwise-disjoint surfaces σ∗0 , ..., σ∗n,each of them dividing W ∗ in two parts, a sequenceof 3-dimensional tiles T ∗i is defined. If we requirethat RK

∗

α (T ∗i ∩ p∗) = T ∗i+1 ∩ q∗ (the condition analo-gous to (1)), the resulting object will be called a solidstack. The tiles can be rearranged by glueing con-gruent faces contained in p∗ and q∗ in order to formthe associated solid strip. We can repeat the argu-ments in Section 1.1 and derive that if the resultingsolid strip is simply connected, its boundary containstwo congruent copies of a surface γ∗. The congruencyrelating them is a rotation around a linear axis C∗parallel to K∗ by the amount corresponding to thedihedral angle between p∗ and q∗. The results in Sec-tion 1.2 also extend to the 3-dimensional case if theconditions are properly adapted.In addition to the refinement possibilities analogous

to the planar case, we can now split the tessellationof a solid stack in order to obtain multiple stacks thatcan be then converted into strips and arranged onenext to the other. To ensure that the tessellation isface-to-face in the strip form, we assume here that thesolid stack is refined only cutting by planes h∗1, ..., h∗lorthogonal to K∗. However, this is not a necessarycondition, as can be appreciated in Figure 1.

3 Discretization

This section studies how polyhedral tiles can be usedto approximate the stacks constructed in previous sec-tions. This can be thought of as simplifying the tilesone would get in the continuous case while preservingtheir stackability and congruency relations.We assume here that a solid stack is given and

that it has been refined by means of the planes (rj0)∗

through K∗ and the planes h∗k orthogonal to K∗.A possible way to discretize the tiles is to substi-tute each separator σ∗i by the lower convex hull ofthe points resulting of its intersection with the raysrjk0 = (rj0)

∗ ∩ h∗k. It is an easy exercise to prove thatthis construction preserves the topology and the con-gruency relations of the tiles. However, the tiles ob-tained in this way may not be convex, a property thatcan be useful in some practical applications. Thisrestriction forces the separators to be planar withineach of the stacks. This is equivalent to approximate

the surface γ∗ by a planar-quadrilateral mesh (PQ-mesh) γ, whose vertices are constrained to lie on pre-scribed rays rjki = RC

∗

iα (rjk0 ). Then, the vertices V jkiof γ can be expressed as V jki = Ojki +λjki u

jki , for some

λjki ∈ R+, where Ojki is the initial point of rjki andujki is its direction. Therefore, the vertices of RC

∗

α (γ)will also lie on these rays and their positions can beretrieved from the variables λjki . However, one shouldensure that the perturbed versions of γ and RC

∗

α (γ)do not intersect. But provided that γ and RC

∗

α (γ) arePQ-meshes, these conditions translate into the simplelinear inequalities λjli < λjli−1.Existing algorithms, like the PQ perturbation de-

tailed in [3], implement the planarity constraints andminimize functions measuring the distance to the sur-face and the quality of the mesh. Since the spe-cific requirements of our method are, as hinted be-fore, translated into linear equations and inequalitiesin the space of coordinates of the vertices, the men-tioned algorithm, based on sequential quadratic pro-gramming [5], can be applied to our construction.The results on 3-space generalize to case where p∗

and q∗ are parallel as well. In this situation, the ver-tices are constrained to lie in rays that have all thesame direction. Therefore, the planarity constraintscan be expressed as linear equations and optimizingwith respect to linear functions can be done very effi-ciently by linear programming.

4 ConclusionWe give the first geometric analysis of the solidsconstructible with the fabrication method CASTon-CAST. We restrict the study imposing some con-straints to the tiles of the tessellation, which leadto simple and geometrically meaningful constructions.We also provide a scheme to approximate the tilesusing polyhedra while preserving their stackability.More general cases can be studied using the intro-duced framework and are left to future work.

References[1] R. Sauer, Differenzengeometrie, Springer, 1970.

[2] A.I. Bobenko, Y.B. Suris, Discrete Differential Ge-ometry: Integrable Structure, Graduate Studies inMathematics Series, Volume 98, AMS, 2008.

[3] Y. Liu, H. Pottmann, J. Wallner, Geometric mod-eling with conical meshes and developable surfaces,ACM Trans. Graphics 25 (2006), 681–689.

[4] Designed by L. Enrique, P. Cepaitis, D. Ordoñez, C.Piles at the Architectural Association School of Ar-chitecture, 2009.

[5] K. Madsen, H. B. Nielsen, O. Tingleff, Optimizationwith Constraints, 2nd ed., 2004.

98

XV Spanish Meeting on Computational Geometry, June 26-28, 2013Improved enumeration of simple topologi al graphs∗Jan Kyn£l† 11Department of Applied Mathemati s and Institute for Theoreti al Computer S ien e, Charles University,Fa ulty of Mathemati s and Physi s, Malostranské nám. 25, 118 00 Praha 1, Cze h Republi Abstra tA simple topologi al graph T = (V (T ), E(T )) is adrawing of a graph in the plane where every two edgeshave at most one ommon point (an endpoint or a rossing) and no three edges pass through a single rossing. Topologi al graphs G and H are isomorphi if H an be obtained from G by a homeomorphism ofthe sphere, and weakly isomorphi if G and H havethe same set of pairs of rossing edges.We generalize results of Pa h and Tóth and the au-thor's previous results on ounting dierent drawingsof a graph under both notions of isomorphism. Weprove that for every graph G with n verti es, m edgesand no isolated verti es the number of weak isomor-phism lasses of simple topologi al graphs that realizeG is at most 2O(n2 log(m/n)), and at most 2O(mn

1/2 logn)if m ≤ n3/2. As a onsequen e we obtain a new up-per bound 2O(n3/2 log n) on the number of interse tiongraphs of n pseudosegments. We improve the upperbound on the number of weak isomorphism lasses ofsimple omplete topologi al graphs with n verti es to2n

2·α(n)O(1) , using an upper bound on the size of a setof permutations with bounded VC-dimension re entlyproved by Cibulka and the author. We show that thenumber of isomorphism lasses of simple topologi algraphs that realize G is at most 2m2+O(mn) and atleast 2Ω(m2) for graphs with m > (6 + ε)n.1 Introdu tion and the resultsA topologi al graph T = (V (T ), E(T )) is a drawing ofa graph G in the plane with the following properties.The verti es of G are represented by a set V (T ) of dis-tin t points in the plane and the edges of G are repre-sented by a set E(T ) of simple urves onne ting the orresponding pairs of points. We all the elements of

∗The full version of the paper is available [8.†Email: kyn lkam.m. uni. z. The author was supportedby the GraDR EUROGIGA GACR proje t No. GIG/11/E023and by the grant SVV-2013-267313 (Dis rete Models and Algo-rithms). Part of the resear h was ondu ted during the Spe ialSemester on Dis rete and Computational Geometry at É olePolyte hnique Féderale de Lausanne, organized and supportedby the CIB (Centre Interfa ultaire Bernoulli) and the SNSF(Swiss National S ien e Foundation).

V (T ) and E(T ) the verti es and the edges of T . Thedrawing has to satisfy the following general position onditions: (1) the edges pass through no verti es ex- ept their endpoints, (2) every two edges have only anite number of interse tion points, (3) every interse -tion point of two edges is either a ommon endpointor a proper rossing (tou hing of the edges is not al-lowed), and (4) no three edges pass through the same rossing. A topologi al graph is simple if every twoedges have at most one ommon point, whi h is eithera ommon endpoint or a rossing. A topologi al graphis omplete if it is a drawing of a omplete graph.We use two dierent notions of isomorphism to enu-merate topologi al graphs.Topologi al graphs G and H are weakly isomorphi if there exists an in iden e preserving one-to-one or-responden e between V (G), E(G) and V (H), E(H)su h that two edges of G ross if and only if the or-responding two edges of H do.Note that every topologi al graph G drawn in theplane indu es a drawing GS2 on the sphere, whi h isobtained by a standard one-point ompa ti ation ofthe plane. Topologi al graphs G and H are isomor-phi if there exists a homeomorphism of the spherewhi h transforms GS2 into HS2 . The isomorphism an be also dened in a ombinatorial way.Unlike the isomorphism, the weak isomorphism an hange the fa es of the involved topologi al graphs,the order of rossings along the edges and also the y li orders of edges around verti es.For ounting the (weak) isomorphism lasses, we onsider all the graphs labeled. That is, ea h vertex isassigned a unique label from the set 1, 2, . . . , n, andwe require the (weak) isomorphism to preserve thelabels. Mostly it makes no signi ant dieren e in theresults as we operate with quantities asymptoti allylarger than n!.For a graphG, let Tw(G) be the number of weak iso-morphism lasses of simple topologi al graphs that re-alize G. Pa h and Tóth [13 and the author [6 provedthe following lower and upper bounds on Tw(Kn).Theorem 1 [6, 13 For the number of weak isomor-phism lasses of simple drawings of Kn, we have2Ω(n2)

≤ Tw(Kn) ≤ ((n− 2)!)n = 2O(n2 logn).

99

Improved enumeration of simple topologi al graphsWe prove generalized upper and lower bounds onTw(G) for all graphs G.Theorem 2 Let G be a graph with n verti es and medges. Then

Tw(G) ≤ 2O(n2 log(m/n)).If m < n3/2, thenTw(G) ≤ 2O(mn

1/2 logn).Let ε > 0. If G is a graph with no isolated verti esand at least one of the onditions m > (1 + ε)n or∆(G) < (1− ε)n is satised, then

Tw(G) ≥ 2Ω(max(m,n logn)).We also improve the upper bound from Theorem 1.Theorem 3 We haveTw(Kn) ≤ 2n

2·α(n)O(1)

.Here α(n) is the inverse of the A kermann fun tion.It is an extremely slowly growing fun tion, whi h anbe dened in the following way [10. α(m) := mink :αk(m) ≤ 3 where αd(m) is the dth fun tion in the in-verse A kermann hierar hy. That is, α1(m) = ⌈m/2⌉,αd(1) = 0 for d ≥ 2 and αd(m) = 1+αd(αd−1(m)) form, d ≥ 2. The onstant in the O(1) notation in the ex-ponent is huge (roughly 430

4), due to a Ramsey-typeargument used in the proof.In the proof of Theorem 3 we use the fa t thatfor simple omplete topologi al graphs, the weak iso-morphism lass is determined by the rotation sys-tem [7, 13. This is ombined with a Ramsey-typetheorem by Pa h, Solymosi and Tóth [12, whi h saysthat a simple omplete topologi al graph with su- iently many verti es ontains a subgraph weakly iso-morphi to a onvex graph or a twisted graph of givensize; see Figure 1. On e we have a onvex graph with5 verti es or a twisted graph with 6 verti es, we mayrestri t the set of possible rotations of other verti esin terms of forbidden subpermutations. The last mainingredient is a re ent ombinatorial result, a slightlysuperexponential upper bound on the size of a set ofpermutations with bounded VC-dimension obtainedtogether with Josef Cibulka [4.The method in the proof of Theorem 2 is more topo-logi al, gives a slightly weaker upper bound, but anbe generalized to all graphs. Here the main tool isa onstru tion of a topologi al spanning tree T of G,whi h is a simply onne ted subset of the single topo-logi al omponent of G ontaining all verti es of Gand satisfying the property that the only nonseparat-ing points of T are the verti es of G. We nd su ha tree onsisting of O(n) onne ted portions of edges

C5

v1 v2 v3 v4 v5 v6

T6Figure 1: The onvex graph C5 and the twisted graphT6.

v1

v2

v4

v3

v5

v6

c1

c2T

v1

c1

1

v2

c2

1

v1

3

v1

4c1

2

v5

c3

2

v2

4

v2

3c3

1

c2

2

v6Figure 2: A topologi al spanning tree T of a simpletopologi al graph with two omponents (left) and the orresponding T -representation (right).of G. By utting the plane along T , we obtain theT -representation of G, whi h is equivalent to a dis with at most 2mn hords, ea h hord orrespondingto a portion of some edge of G. See Figure 2. Wegive an upper bound on the number of inequivalentT -representations, exploiting the fa t that many por-tions of edges do not ross.We further generalize Theorem 3 by removing al-most all topologi al aspe ts of the proof. The re-sulting theorem is a purely ombinatorial statement,involving n-tuples of y li permutations avoiding a ertain simple substru ture.We also onsider the lass of simple omplete topo-logi al graphs with maximum number of rossings andsuggest an alternative method for obtaining an upperbound on the number of weak isomorphism lasses ofsu h drawings.An arrangement of pseudosegments (or also 1-strings) is a set of simple urves in the plane su h thatany two of the urves ross at most on e. An inter-se tion graph of pseudosegments (also alled a stringgraph of rank 1) is a graph G su h that there exists anarrangement of pseudosegments with one pseudoseg-ment for ea h vertex of G and a pair of pseudoseg-ments rossing if and only if the orresponding pairof verti es forms an edge in G. Using tools from ex-tremal graph theory, Pa h and Tóth [13 proved thatthe number of interse tion graphs of n pseudoseg-ments is 2o(n

2). As a spe ial ase of Theorem 2 we100

XV Spanish Meeting on Computational Geometry, June 26-28, 2013obtain the following upper bound.Theorem 4 There are at most 2O(n3/2 logn) interse -tion graphs of n pseudosegments.The best known lower bound for the number of (un-labeled) interse tion graphs of n pseudosegments is2Ω(n logn). This follows by a simple onstru tion orfrom the the fa t that there are 2Θ(n logn) nonisomor-phi permutation graphs with n verti es.Let T (G) be the number of isomorphism lasses ofsimple topologi al graphs that realize G. The follow-ing theorem generalizes the result T (Kn) = 2Θ(n4)from [7.Theorem 5 Let G be a graph with n verti es, medges and no isolated verti es. Then T (G) ≤

2m2+O(mn). More pre isely,T (G) ≤

(6mn

2mn

)(m2 + 6mnm2

2 + 2mn

)

· 2O(n log n)

≤ 2m2+2mn(1+3 log2 3)+O(n logn), and

T (G) ≤ 2m2+4mn

·

(2mn+ m

2

2

2mn

)

· 2O(n logn)

≤ 2m2+2mn(log(1+ m

4n )+2+log2 e)+O(n logn).Let ε > 0. For graphs G with m > (6 + ε)n we haveT (G) ≥ 2Ω(m2).For graphs G with m > ω(n) we have

T (G) ≥ 2m2/60

− o(1).The two upper bounds on T (G) ome from two es-sentially dierent approa hes to enumerating isomor-phism lasses of T -representations. In the rst ap-proa h, we redu e the problem to enumerating sim-ple quadrangulations of the dis [9. In the se ondapproa h, we split the problem into two parts: enu-merating hord diagrams [14 and enumerating iso-morphism lasses of arrangements of pseudo hords.The rst method gives better asymptoti results fordense graphs, whereas the se ond one is better forsparse graphs (roughly, with at most 35n edges). Forgraphs with m = O(n) the se ond term in the expo-nent be omes more signi ant. Sin e m ≥ n/2, theexponent in the rst upper bound an be bounded by23.118m2 + o(1), using the entropy bound for the bi-nomial oe ient. Similarly, the exponent in the se -ond upper bound an be bounded by 11.265m2+o(1).For su h very sparse graphs (for example, mat hings),however, better upper bounds an be dedu ed moredire tly from other known results.The upper bound T (G) ≤ 2O(m2) is trivially ob-tained from the upper bound on the number of unla-beled plane graphs (or planar maps). Indeed, every

drawing G of G an be transformed into a plane graphH by subdividing the edges of G by its rossings andregarding the rossings of G as new 4-valent verti esin H . The graph H has thus at most n +

(m

2

) ver-ti es, at most m+2(m

2

)= m2 edges, no loops and nomultiple edges. Tutte [17 showed that there are

2(2M)!3M

M !(M + 2)!= 2(log2(12)+o(1))Mrooted onne ted planar maps with M edges (seealso [2, 3, 5). Walsh and Lehman [18 showed thatthe number of rooted onne ted planar loopless mapswith M edges is

6(4M + 1)!

M !(3M + 3)!= 2(log2(256/27)+o(1))M .This implies the upper bound T (G) ≤

2(log2(256/27)+o(1))m2 . Somewhat better estimates ould be obtained by redu ing the problem to ount-ing 4-regular planar maps [15, 16, sin e typi allyalmost all verti es in H are the 4-valent verti es ob-tained from the rossings of G. But su h a redu tionwould be less straightforward and the resulting upperbound 2(12 log2(196/27)+o(1))m2 still relatively high fordense graphs (for graphs with more than 27n edgesthe two upper bounds from Theorem 5 are better).The proof in [7 implies the upper bound T (Kn) ≤

2(1/12+o(1))(n4), although it is not expli itly statedthere. However, the key Proposition 7 in [7 is in- orre t. We prove a orre t version in the full paper.Note that by the redu tion to ounting planarmaps, for every xed onstant k, we also obtain theupper bound 2O(km2) on the number of isomorphism lasses of onne ted topologi al graphs with m edgeswhere all pairs of edges are allowed to ross k times.2 A few open problemsThe problem of ounting the asymptoti number ofnonequivalent simple drawings of a graph in theplane is answered only partially. Many open ques-tions remain.The gap between the lower and upper bounds onTw(G) proved in Theorem 2 is wide open, espe iallyfor graphs with low density. For graphs with cn2edges, the lower and upper bounds on logTw(G) dif-fer by a logarithmi fa tor. We onje ture that the orre t answer is loser to the lower bound.We do not even know whether Tw(G) is a monotonefun tion with respe t to the subgraph relation, sin ethere are simple topologi al graphs that annot be ex-tended to simple omplete topologi al graphs. Due tosomewhat rigid properties of simple omplete topo-logi al graphs, we have a mu h better upper boundfor the omplete graph than, say, for the ompletebipartite graph on the same number of verti es.

101

Improved enumeration of simple topologi al graphsProblem 1 Does the omplete graph Kn maximizethe value Tw(G) among the graphs G with n verti es?More generally, is it true that Tw(H) ≤ Tw(G) if H ⊆

G?Our methods for proving upper bounds on the num-ber of weak isomorphism lasses of simple topologi algraphs do not generalize to the ase of topologi algraphs with two rossings per pair of edges allowed.Problem 2 What is the number of weak isomor-phism lasses of drawings of a graph G where everytwo independent edges are allowed to ross at mosttwi e and every two adja ent edges at most on e?For the omplete graph with n verti es, Pa h andTóth [13 proved the lower bound 2Ω(n2 logn) and theupper bound 2o(n4).A nontrivial lower bound an be proved also in the ase when G is a mat hing. A kerman et al. [1 on-stru ted a system of n x-monotone urves where ev-ery pair of urves interse t in at most one point wherethey either ross or tou h, with Ω(n4/3) pairs of tou h-ing urves. Eyal A kerman (personal ommuni ation)noted that this also follows from an earlier result byPa h and Sharir [11, who onstru ted an arrange-ment of n segments with Ω(n4/3) verti ally visiblepairs of disjoint segments. By hanging the drawing inthe neighborhood of every tou hing point, we obtain

2Ω(n4/3) dierent interse tion graphs of 2-interse ting urves, also alled string graphs of rank 2 [13. Thisimproves the trivial lower bound observed by Pa hand Tóth [13.A knowledgementsThe author thanks Josef Cibulka for dis ussions aboutenumerating various ombinatorial obje ts.Referen es[1 E. A kerman, R. Pin hasi and S. Zerbib, Ontou hing urves, Bernoulli Reunion Confer-en e on Dis rete and Computational Geometry ,EPFL, Lausanne, 2012.[2 E. A. Bender and L. B. Ri hmond, A survey ofthe asymptoti behaviour of maps, Journal ofCombinatorial Theory, Series B 40(3) (1986),297329.[3 E. A. Bender and N. C. Wormald, The numberof loopless planar maps, Dis rete Mathemati s54(2) (1985), 235237.[4 J. Cibulka and J. Kyn£l, Tight bounds on themaximum size of a set of permutations with

bounded VC-dimension, Journal of Combinato-rial Theory, Series A 119(7) (2012), 14611478.[5 M. Drmota and M. Noy, Universal exponentsand tail estimates in the enumeration of planarmaps, Ele troni Notes in Dis rete Mathemati s38 (2011), 309317.[6 J. Kyn£l, Crossings in topologi al graphs, masterthesis, Charles University, Prague, 2006.[7 J. Kyn£l, Enumeration of simple omplete topo-logi al graphs, European Journal of Combina-tori s 30(7) (2009), 16761685.[8 J. Kyn£l, Improved enumeration of simple topo-logi al graphs, submitted. Manus ript availableat arXiv:1212.2950.[9 R.C. Mullin and P.J. S hellenberg, The enumer-ation of -nets via quadrangulations, Journal ofCombinatorial Theory 4(3) (1968), 259276.[10 G. Nivas h, Improved bounds and new te h-niques for DavenportS hinzel sequen es andtheir generalizations, Journal of the ACM 57(3)(2010), 144.[11 J. Pa h and M. Sharir, On verti al visibility in ar-rangements of segments and the queue size in theBentley-Ottmann line sweeping algorithm, SIAMJournal on Computing 20(3) (1991), 460470.[12 J. Pa h, J. Solymosi and G. Tóth, Un-avoidable ongurations in omplete topologi algraphs, Dis rete and Computational Geometry30 (2003), 311320.[13 J. Pa h and G. Tóth, How many ways anone draw a graph?, Combinatori a 26(5) (2006),559576.[14 R. C. Read, The hord interse tion problem,Annals of the New York A ademy of S ien es319(1) (1979), 444454.[15 H. Ren and Y. Liu, Enumerating near-4-regularmaps on the sphere and the torus, Dis rete Ap-plied Mathemati s 110(23) (2001), 273288.[16 H. Ren, Y. Liu and Z. Li, Enumeration of2- onne ted Loopless 4-regular maps on theplane, European Journal of Combinatori s 23(1)(2002), 93111.[17 W. T. Tutte, A ensus of planar maps, CanadianJournal of Mathemati s 15 (1963), 249271.[18 T. R. S. Walsh and A. B. Lehman, Countingrooted maps by genus III: Nonseparable maps,Journal of Combinatorial Theory, Series B 18(1975), 222259.102

http://arxiv.org/abs/1212.2950

XV Spanish Meeting on Computational Geometry, June 26-28, 2013On three parameters of invisibility graphs ∗Josef Cibulka†1, Miroslav Korbelá°‡2, Jan Kyn£l§1, Viola Mészáros¶3, Rudolf Stola°‖1, and Pavel Valtr∗∗11Department of Applied Mathemati s and Institute for Theoreti al Computer S ien e, Charles University,Fa ulty of Mathemati s and Physi s, Malostranské nám. 25, 118 00 Prague, Cze h Republi 2Department of Mathemati s and Statisti s, Fa ulty of S ien e, Masaryk University, Kotlá°ská 2, 611 37 Brno,Cze h Republi 3Institute for Mathemati s, Te hni al University of Berlin, Strasse des 17. Juni 136, 10623 Berlin, GermanyAbstra tThe invisibility graph I(X) of a set X ⊆ Rd is a (pos-sibly innite) graph whose verti es are the points ofX and two verti es are onne ted by an edge if andonly if the straight-line segment onne ting the two orresponding points is not fully ontained in X . We onsider the following three parameters of a set X :the lique number ω(I(X)), the hromati numberχ(I(X)) and the minimum number γ(X) of onvexsubsets of X that over X .We settle a onje ture of Matou²ek and Valtr laim-ing that for every planar set X , γ(X) an be boundedin terms of χ(I(X)). As a part of the proof we showthat a dis with n one-point holes near its boundaryhas χ(I(X)) ≥ log log(n) but ω(I(X)) = 3.We also nd sets X in R5 with χ(I(X)) = 2, butγ(X) arbitrarily large.1 Introdu tionLet X be a subset of a d-dimensional Eu lidean spa e.We say that two points x, y ∈ X see ea h other ifthe straight-line segment xy onne ting x and y isa subset of X . The invisibility graph I(X) of a set

∗This resear h was supported by the proje t CE-ITI(GACR P202/12/G061) of the Cze h S ien e Foundation andby the grant SVV-2013-267313 (Dis rete Models and Algo-rithms). The se ond author was supported by the proje tCZ.1.07/2.3.00/20.0003 of the Operational Programme Edu a-tion for Competitiveness of the Ministry of Edu ation, Youthand Sports of the Cze h Republi . The fourth author was alsopartially supported by ESF EuroGiga proje t ComPoSe (IP03),by OTKA Grant K76099 and by OTKA Grant 102029. Therst, the third and the sixth author were partially supported byproje t GAUK 52410. Part of the resear h was ondu ted dur-ing the Spe ial Semester on Dis rete and Computational Geom-etry at É ole Polyte hnique Féderale de Lausanne, organizedand supported by the CIB (Centre Interfa ultaire Bernoulli)and the SNSF (Swiss National S ien e Foundation).†Email: ibulkakam.m. uni. z.‡Email: miroslav.korbelargmail. om.§Email: kyn lkam.m. uni. z.¶Email: meszarosmath.tu-berlin.de.‖Email: rudakam.m. uni. z.

∗∗Email: valtrkam.m. uni. z.

X ⊆ Rd is a graph whose verti es are the points ofX and two verti es are onne ted by an edge if andonly if they do not see ea h other. Let χ(G) be the hromati number of a graph G and let ω(G) be its lique number. For a set X ⊆ Rd we dene γ(X) tobe the minimum possible number of onvex subsets ofX that over X . Further, let χ(X) := χ(I(X)) andω(X) := ω(I(X)).Sets X with ω(X) = n − 1 are sometimes alledn- onvex [8.Observe that ω(X) ≤ χ(X) ≤ γ(X) for any set X .If a planar set X is losed, then γ(X) an bebounded by a fun tion of ω(X). This was provedby Breen and Kay [2 and the urrent best knownupper bound is γ(X) ≤ O(ω(X)3) by Matou²ek andValtr [6. From the other dire tion, there exist ex-amples by Matou²ek and Valtr [6 with γ(X) ≥

Ω(ω(X)2).However, if we don't restri t ourselves to losedsets, there is no upper bound on γ(X) even for setswith ω(X) = 3. An example is the dis Dλ with λone-point holes pun tured in the verti es of a regu-lar onvex λ-gon near the boundary of Dλ, for whi hω(Dλ) = 3, but γ(Dλ) = ⌈λ/2⌉+ 1 (see [6).A one-point hole in a set X ⊂ Rd is a point thatforms a path- onne ted omponent of Rd

\ X . Letλ(X) be the number of one-point holes in the set X .The example of the set Dλ led to studying the prop-erties of planar sets with a limited number of one-point holes by Matou²ek and Valtr [6. In parti ular,they proved the following theorem.Theorem 1 (Matou²ek and Valtr [6) Let X ⊆

R2 be a set with ω(X) = ω < ∞ and λ(X) = λ < ∞.Thenγ(X) ≤ O(ω4 + λω2).For any ω ≥ 3 and λ ≥ 0 they also found sets Xwith ω(X) = ω, λ(X) = λ and γ(X) ≥ Ω(ω3 + ωλ).Matou²ek and Valtr [6 onje tured that for an ar-bitrary planar set X , the value of γ(X) is bounded bya fun tion of χ(X). Then χ(X) annot be bounded

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On three parameters of invisibility graphsby a fun tion of ω(X) as shows the above examplewith Dλ.Lawren e and Morris [4 proved that for every kthere exists n0(k) su h that whenever S is a set ofnitely many points in the plane and |S| ≥ n0(k),then χ(R2\ S) ≥ k. 1 Thus, whenever X is the om-plement of a nite set of points, λ(X) an be boundedin terms of χ(X). This implies, by Theorem 1, thatthe value of γ(X) an be bounded in terms of χ(X),settling the onje ture of Matou²ek and Valtr in thespe ial ase when X is a omplement of a nite set ofpoints.In this paper, we strengthen the result of Lawren eand Morris [4 and settle the onje ture for every pla-nar set X .The tower fun tion Tl(k) is dened re ursively as

T0(k) = k and Th(k) = 2Th−1(k). Its inverse is theiterated logarithm log(l)(n), that is, log(0)(n) = n andlog(l)(n) = log(log(l−1)(n)).Theorem 2 Any set X ⊆ R2 with χ(X) = χ < ∞satises

γ(X) ≤ O(24T2(χ)· χ3).The proof of Theorem 2 is omitted from this ex-tended abstra t. In the full version of the paper, wealso show that for every dimension d, λ(X) an bebounded in terms of χ(X) for sets X ⊂ Rd. Thisanswers Question 6 of Lawren e and Morris [4.A set X is star-shaped if X ontains a point x ∈ Xthat sees every other point of X .In Se tions 2 and 3 we show that χ and γ an beseparated in dimensions 5 and more.Theorem 3 For every positive integer g there existstar-shaped sets1. X ⊂ R6 satisfying χ(X) = 2 and γ(X) ≥ g and2. Xc ⊂ R6 that is losed and satises χ(Xc) = 4and γ(Xc) ≥ g.Theorem 4 For every positive integer g there existstar-shaped sets1. X ⊂ R5 satisfying χ(X) = 2 and γ(X) ≥ g and2. Xc ⊂ R5 that is losed and satises χ(Xc) = 6and γ(Xc) ≥ g.Problem 1 Does there exist a fun tion f su h that

γ(X) ≤ f(χ(X)),1. for every set X ⊆ R3?2. for every set X ⊆ R4?All logarithms in this paper are binary. We usethe notation xy for the straight line segment betweenpoints x and y.1The graph GS in the paper of Lawren e and Morris is pre- isely the invisibility graph of R2 \ S.

2 Constru tions in dimension 62.1 Set with hromati number 2We prove part 1 of Theorem 3. Part 2 is omitted fromthis extended abstra t.Let Pn be the y li polytope on n ≥ 7 verti es(see for example [5) and Vn its set of verti es. Thusthe onvex hull of every triple of points from Vn is atriangular fa e of Pn.Lemma 5 Let Kn be the omplete graph on the setV of n ≥ 7 verti es and let k := ⌈2 log(n) + 2⌉. Itis possible to orient the edges of Kn so that every setV ′

⊆ V of size at least k ontains a dire ted triangle.Proof. For brevity, we all a set V ′ good if it ontainsa dire ted triangle.We orient the edges randomly and show that withpositive probability, every set V ′⊆ V of size at least

k is good.First, we will bound the probability bk that a givenset V ′ of k verti es is bad.If there exists a dire ted y le on V ′ of lengthgreater than 3, then one of the two y les reatedby adding an arbitrary diagonal to the y le is againdire ted. Thus there exists a dire ted triangle on V ′.There are 2k(k−1)/2 possible orientations of edgesof a omplete graph on k verti es out of whi h k! area y li . Thusbk =

k!

2k(k−1)/2= k!2−k2/2+k/2.The probability that some k-tuple V ′ of verti es isbad is thus at most

(n

k

)

bk ≤

nk

k!bk = 2k log(n)−k2/2+k/2 =

= 2k(log(n)−k/2+1/2)≤

≤ 2k(log(n)−log(n)−1+1/2)≤ 2k(−1/2) < 1.

We x the orientation of the edges of Pn in whi hevery k-tuple of verti es is good. A triangular fa eof Pn has dire ted boundary if the three edges of thefa e form a dire ted y le. The set X is onstru tedby pun turing a one-point hole in the bary enter oftriangular fa es of Pn with dire ted boundary.Verti es of Pn are olored bla k. Edges are ut inthirds. In every edge, the interior of the middle thirdtogether with the point at one third loser to the endof the edge is olored white. The rest of the edges isbla k. The oloring of triangular fa es with dire tedboundary is depi ted on Fig. 1. The rest of X is olored bla k.All the edges of the invisibility graph of X are be-tween pairs of points lying on the same triangular fa e104

XV Spanish Meeting on Computational Geometry, June 26-28, 2013vi

vj

vk

p1 p2

p3

p4p5

p6

Figure 1: Coloring of a fa e with dire ted bound-ary. Lines determined by pairs (p1, p4), (p2, p5) and(p3, p6) interse t in the bary enter and split the trian-gle into mono hromati regions. Full lines and grayareas represent bla k olor, the rest is white.with dire ted boundary. The oloring is proper onea h of these fa es and thus the 2- oloring of the wholeX is proper.If a onvex set C ontains at least k verti es of X ,then it ontains a triangular fa e with dire ted bound-ary and thus C ontains a one-point hole. Therefore

γ(X) ≥n

2 log(n) + 3.3 Constru tions in dimension 5Here we present some parts of the proof of part 1 ofTheorem 4. The rest of the proof is omitted from thisextended abstra t.The onstru tions are similar to those in dimension

6: for part (1) of the theorem, the set X is a losed y li polytope with one-point holes in some of the2-dimensional fa es. For part (2), instead of pointswe remove small 5-dimensional polytopes atta hed tothe 2-dimensional fa es. The dieren e from the on-stru tion in dimension 6 is in the pla ement of theholes: here we annot apply the same argument as inthe previous se tion sin e for the y li polytope in di-mension 5 only quadrati ally many triples of verti esindu e a 2-dimensional fa e and there is a 2- oloringof the vertex set in whi h no 2-dimensional fa e ismono hromati .In the full version of the paper, we show two dif-ferent ways how to hoose the holes. In the rst on-stru tion we essentially show that randomly hosenholes will do, but the proof (interestingly) requires arather nontrivial result from group theory. Also the onstru tion proves only part (1) of the theorem. Inthe se ond onstru tion we spe ify the lo ations of theholes pre isely. Moreover, we show that the holes anbe enlarged to open pyramids, whi h shows part (2)of the theorem. The proof of part (2) of the theoremis analogous to the proof of part (2) of Theorem 3 andis omitted from this abstra t.Let Pn be the 5-dimensional y li polytope onn ≥ 6 verti es with (ordered) vertex set Vn =

v1, v2, . . . , vn. For brevity, we all the triangularfa e with verti es vi, vj and vk the ijk triangle. Sim-ilarly, the ij edge is the edge between verti es vi andvj . The 2-dimensional fa es of Pn are the triangles• 1ij for every 1 < i < j ≤ n (type 1ij triangles),• ijn for every 1 ≤ i < j < n (type ijn triangles),• i(i+1)j for every 1 < i < j−1 < n (type i(i+1)jtriangles) and• ij(j+1) for every 1 < i < j < n−1 (type ij(j+1)triangles).3.1 Covering with onvex setsIn the onstru tions proving part (1) of Theorem 4, weremove a one-point hole from every type 1ij triangle.In the onstru tion proving part (2), we remove anopen at simplex instead of the point (as in Se tion 2).The following lemma shows that in both ases, theresulting set an not be overed by a bounded numberof onvex sets.Lemma 6 Let X be a subset of Pn su h that ev-ery edge of Pn is a subset of X and none of thetype 1ij triangles is a subset of X. Then γ(X) ≥

Ω(logn/ log logn).Proof. Let X = C1 ∪ C2 ∪ · · · ∪ Ck be a overing ofX with onvex subsets of X . The overing indu esa partition of ea h open edge 1i, 2 ≤ i ≤ n, intoat most k intervals I1i , I

2i , . . . , I

ki

i , where ea h of theintervals Iji is overed by one of the onvex sets Cl(i,j).Sin e the onvex sets in the overing may overlap, thispartition need not be unique; in su h a ase we justpi k one.We say that the partitions of two edges 1i and 1i′are of the same type if ki = ki′ , l(i, p) = l(i′, p) forea h p = 1, 2, . . . , ki (in other words, the olors ap-pear in the same order along the edges), and for ea hp = 1, 2, . . . , ki the type of the interval Ipi (that is, losed, open, or half- losed from the left/right) is thesame as the type of the interval Ipi′ . Degenerate one-point intervals are onsidered as losed. The numberof types of the partitions is at most 2k · k! · 2k−1. In-deed, there are at most 2k subsets of olors, ea h ofthe subsets an be linearly ordered in at most k! ways,and there are at most k − 1 boundary points sharedby two intervals, where one of the intervals is lo ally losed and the other one lo ally open.It follows that if n > 2k · k! · 2k−1 + 1, then thereare two edges 1i and 1i′ of the same type. The onvexhulls conv(Ipi ∪ Ipi′ ) over the whole open triangle 1ii′,in luding the one-point hole inside, whi h is a ontra-di tion. Therefore n ≤ 2k · k! · 2k−1+1, whi h impliesthat γ(X) ≥ Ω(logn/ log logn).

105

On three parameters of invisibility graphs4 Con luding remarksTo solve Problem 1 in dimension 4 we ould use a onstru tion similar to those in dimensions 5 and 6provided the following problem has a positive answer.Problem 2 Does there exist for every k a onvexsimpli ial polytope P (k) in R4 su h that in every ol-oring of verti es of P (k) by k olors we an nd atriangular fa e whose verti es are mono hromati ?Assuming the polytope P (k) from Problem 2 exists,the set X from Problem 1 is obtained from P (k) bymaking a one-point hole in an arbitrary point insideevery triangular fa e. Su h a set X annot be overedby k onvex sets sin e otherwise one of the onvex setswould ontain three verti es of a triangular fa e.The invisibility graph I(X) an be olored by 13 olors in the following way. All the verti es of P (k)get olor 1. Tan er [7 has shown that the edges of ev-ery 2-dimensional simpli ial omplex PL-embeddablein R3 an be olored by 12 olors so that for everytriangular fa e, the three edges on its boundary havethree dierent olors. This applies, in parti ular, tothe 2-skeleton of every 4-dimensional onvex simpli- ial polytope. We use olors 2, 3, . . . , 13 to olor theinteriors of edges of P (k) in this way. For ea h tri-angular fa e and ea h point p on its boundary, theinterior of the segment one ting the one-point holewith p is olored by the olor of p. All the remain-ing points of X are isolated in I(X) and thus may be olored arbitrarily.The boundary omplex of a 4-dimensional onvexsimpli ial polytope is a spe ial ase of a triangula-tion of S3. If we relax the ondition on polytopalityin Problem 2 and ask only for a triangulation of S3,then the answer is yes. Heise et al. [3 onstru ted,for every k, a 2-dimensional simpli ial omplex lin-early embedded in R3 su h that in every oloring ofits verti es with k olors at least one of the trian-gles is mono hromati . We found the same simpli ial omplex independently, modifying Boris Bukh's on-stru tion, whi h was ommuni ated to us by MartinTan er. The verti es of the omplex are pla ed onthe moment urve and a suitable non rossing subsetof triangles is hosen for the fa es. It remains to ex-tend the embedded omplex to a triangulation of thewhole R3, or S3 [1.Referen es[1 K. Adiprasito, F. Lutz and J. Moller, unpub-lished manus ript.[2 M. Breen and D. C. Kay, General de ompositiontheorems for m- onvex sets in the plane, IsraelJournal of Mathemati s 24 (1976), 217233.

[3 C. G. Heise, K. Panagiotou, O. Pikhurko and A.Taraz, Coloring d-embeddable k-uniform hyper-graphs, arXiv:1209.4879 (2012).[4 J. Lawren e and W. Morris, Finite sets as om-plements of nite unions of onvex sets, Dis- rete & Computational Geometry 42 (2009), no.2, 206218.[5 J. Matou²ek, Le tures on Dis rete Geometry,Springer-Verlag, New York (2002).[6 J. Matou²ek and P. Valtr, On visibility and over-ing by onvex sets, Israel Journal of Mathemati s113 (1999), no. 3, 341379.[7 M. Tan er, unpublished manus ript.[8 F. Valentine, A three point onvexity property,Pa i Journal of Mathemati s 7 (1957), no. 2,12271235.

106



César Hernández–Vélez∗1 and Jesús Leaños†2

1Instituto de Física, Universidad Autónoma de San Luis Potosí, San Luis Potosí, México 780002Unidad Académica de Matemáticas, Universidad Autónoma de Zacatecas, Zacatecas, México 98060

Abstract

A graph is crossing-critical if its crossing number de-creases when we remove any of its edges. Recentlyit was proved that if a non-planar graph G is ob-tained by adding an edge to a cubic polyhedral (pla-nar 3-connected) graph, then G can be made crossing-critical by a suitable multiplication of its edges. Herewe show: (i) a new family of graphs that can be trans-formed into crossing-critical graphs by a suitable mul-tiplication of its edges, and (ii) a family of graphs thatcannot be made crossing-critical by any multiplicationof its edges.

Introduction

This work is motivated by the following question set-tled by Beaudou et al. in [1]: to what extent iscrossing-criticality a property that is inherent to thestructure of a graph, and to what extent can it beinduced on a non crossing-critical graph by multiply-ing (all or some of) its edges? In [1] a family of noncrossing-critical graphs are transformed into crossing-critical graphs by multiplying (adding parallel) edges.The use of parallel edges has been essential in many

other important results on crossing number. For ex-ample, in [2] was proved that for every a > b > 0,there exist a graph G with crossing number a in theplane, crossing number b in the torus, and crossingnumber 0 in the double torus. This is called the ori-entable crossing sequence of a graph G. In [4] wasreported a conjecture of R. B. Richter which statesthat crossing-critical graphs have bounded maximumdegree. This conjecture was disproved by Dvořák andMohar [3], who exhibited crossing-critical graphs withlarge maximum degree. In all these papers the use ofweighted (also called “thick”) edges is essential.No simple graph that satisfies the properties as de-

fined in [2] and [3] are not known. On the otherhand there exist simple non crossing-critical graphssuch that if we multiply its edges (or equivalently,assign weights on them) then they become crossing-critical, as in [1]. Thus, important questions remain:

∗Email: [email protected].†Email: [email protected].

What makes a graph crossing-critical? Does everynon-planar graph become crossing-critical by a weightassignment to its edge set?In this paper we give two infinite families G andG′ of graphs such that (i) every graph in G remainsnon crossing-critical even after any multiplications ofits edges, and (ii) every graph in G′ is not crossing-critical, but after a suitable multiplication of its edges,it is transformed into a crossing-critical graph. Let usproceed to define these families.Let Wn be the wheel with n + 1 vertices, n ≥ 5,

and let v0 be the degree n vertex. The remaining nvertices are labeled v1, v2, . . . , vn in the order in whichthey appear in the n-cycle. We add a new vertex uto Wn which is joined to vertices v0, v1 and v3 anddenote by Gn the resulting graph. Let G′n be thegraph that results by removing the edges v0v1 andv0v3 from Gn. We define G := Gn : n ≥ 5 andG′ := G′n : n ≥ 5. See Figure 1. Note that eachgraph of G ∪ G′ is 3-connected.It is not difficult to find non-planar graphs which

cannot be made crossing-critical by any multiplica-tion of its edges. For example, if G is a non-planargraph and e is a cut edge of G, then G cannot be madecrossing-critical by any multiplication of its edges be-cause e cannot be made critical. Thus, some con-nectivity assumption is needed in order to guaranteethat a non-planar graph can be made crossing-critical.In [1], for example, the graphs under consideration areassumed to be internally 3–connected and such an as-sumption plays a central role in its work. On theother hand, as we will see in Theorem 1, the familyG is interesting because shows that 3-connectednessproperty by itself is not sufficient to ensure that anon-planar graph can be made crossing-critical.Our main results are the following.

Theorem 1 Any graph Gn ∈ G is not crossing-critical. Moreover, Gn cannot be made crossing-critical by any multiplication of its edges.

Theorem 2 Every graph G′n ∈ G′ is not crossing-critical, but there exists a suitable multiplication ofits edges such that the resulting graph G′n is crossing-critical.

107


An edge e of a graph G is a Kuratowski edge if thereis a subgraph H of G which is isomorphic to a subdi-vision of a K5 or K3,3 and e lies on H. In [1] was con-jectured that a graph whose edges are all Kuratowskibecomes crossing-critical after a suitable multiplica-tion of its edges. In this sense, note that the graphsin G′ are consistent with this conjecture because anyedge in G′n is a Kuratowski edge (unlike the edgesv0v1 and v0v3 in Gn).

1 Multigraphs andweighted graphs

We recall that the crossing number cr(G) of a graphG is the minimum number of pairwise intersections ofedges in a drawing of G in the plane. An edge e of G iscrossing-critical if cr(G− e) < cr(G), and say that Gis crossing-critical if all its edges are crossing-critical.Recall that a weighted graph is a pair (G,w) where

G is a graph and w is a function (the weight assign-ment) that assigns to each edge e of G a number w(e)(the weight). The weight assignment is positive (re-spectively, integer) if w(e) is a positive (respectively,integer) number for any edge e of G. We only considerpositive integer weight assignments.We extend the concept of crossing number to

weighted graphs (G,w) in an analogous way, just tak-ing to account that a crossing between the edges e ande′ contributes w(e) · w(e′) to cr (G,w). A drawing Dof (G,w) is optimal if cr(D) = cr(G,w).We now proceed to define what a crossing-critical

edge is in a weighted graph. Let (G,w) be a weightedgraph, an edge e of (G,w) is crossing-critical ifcr (G,we) < cr (G,w), where the weight assignmentwe is defined by,

we(f) =

w(f) if f 6= e,

w(f)− 1 if f = e.

As usual, (G,w) is crossing-critical if all its edgesare crossing-critical.Let G∗ be a multigraph, and let G be its underly-

ing simple graph. We define the associated weightedgraph (G,w∗) of G∗ as follows: for every edge e of G,we define w∗(e) as the multiplicity of e in G∗. Thefollowing observation is straightforward.

Remark 1 Let G∗ be a multigraph and let (G,w∗) beits associated weighted graph. Then G∗ is crossing-critical if and only if (G,w∗) is crossing-critical.

2 Proofs of Theorems 1 and 2

For brevity, let (i) α0 := v0u and αi := v0vi, i =1, . . . , n; (ii) βi := vivi+1, i = 1, . . . , n with vn+1 = v1;

v6 v5

v4

v3v2

v1

u

v0

β2

β3

β5

β4

β1

β6

α3

α4

α0

α2

α5α6

α1

γ2

γ1

v6 v5

v4

v3v2

v1

u

v0

β2

β3

β4

β1

β6

β5

α4

α0

α2

α5α6

γ2

γ1

Figure 1: A drawing D of G6 (above), and a drawingD′ of G′6 (below).

and (iii) γ1 := uv1 and γ2 := uv3. Under this nota-tion, observe that G′n = Gn −α1, α3. See Figure 1.

Lemma 3 Let w′ be a positive integer weight assign-ment on G′n. Then there exists an optimal drawing of(G′n, w

′) in which we can add both edges α1 and α3

without increasing the crossing number.

Proof. Let D′ be an optimal drawing of (G′n, w′).

We divide the proof according to the edges that areinvolved in a crossing in D′.

(A) Suppose that α0 is crossed in D′ (analogously forα2).

Since D′ is an optimal drawing, α0 does not crosswith adjacent edges, therefore α0 crosses with a β-edge, say βj . Let D∗ be the drawing defined as fol-lows. Draw a simple regular n-polygon such thatv1, v2, . . . , vn (in that order) are its vertices, place v0in the center of such a polygon and add the αi-edge,i = 2, 4, 5, . . . , n, as straight line segments. Now drawα0 as an straight line segment crossing the edge βj .Finally the edges γ1 and γ2 can be added around then-polygon boundary without introducing new cross-ings. So D∗ has just one crossing and thereforecr(D∗) ≤ cr(D′). But since D′ is an optimal drawing,we must have that cr(D∗) = cr(G′, w′). Moreover, wecan add the edges α0 and α3 as straight line segmentswithout adding a new crossing, as required.

(B) Both edges in any of β1, β2, γ1, γ2, β1, γ2,γ1, β2 are not crossed in D′.

108


v6

R

S

Q

v4

v5

v1

v3

v2v0u

β2

β6

β4

β3

β1

β5

α2α0

α5

α4

α6

γ2

γ1

Figure 2: Drawing of G′6 where γ1, γ2, β1, β2, α0 andα2 form a plane K2,3

We only analyze the case in which both edges ofβ1, γ2 are not crossed in D′ (the rest of the cases canbe verified similarly). By (A) we can assume that nei-ther α0 nor α2 are crossed in D′. Because none of β1and γ2 has a cross in D′, we can add α1 (respectivelyα3) following the uncrossed path α2β1 (respectivelyα0γ2) without introducing new crossings.

(C) Suppose β1 crosses γ2 in D′ (analogously, β2crosses γ1).

Draw a plane wheel with vertices v1, . . . , vn forminga n-cycle and the vertex v0 as the center. All edges αi,i 6= 1, 3 are the spokes of this wheel. Then draw thepath α0γ1 as a spoke. Finally add the edge γ2 onlycrossing the edge β1. Since the number of crossingsof such a drawing is less than or equal to the numberof crossings of D′, it is optimal. In this new drawingwe can add α1 and α3 without increasing the crossingnumber.By (A) and (C) we can assume that neither γ1,

γ2, β1, β2, α0 nor α2 cross each other. Thus γ1, γ2,β1, β2, α0 and α2 form a plane K2,3. Without anyloss of generality, we may assume that v0 is in thebounded region defined by the cycle γ1β1β2γ2. LetR,S and Q denote the disjoint regions bounded bythe cycles α0α2β1γ1, α0γ2β2α2 and γ1γ2β2β1, respec-tively. (See Figure 2)Let P be the path β3β4 . . . βn. Let µ, ν, and ρ be

the element of β1, γ1, β2, γ2, and β1, β2, γ1, γ2with less weight, respectively.Now we proceed according to the interior vertices

of P that are in each of R,S and Q.

(D) No interior vertices of P are in Q.

Then, all the interior vertices of P are in R, S, orboth of them.(D.i) All interior vertices of P are in R (analogouslyfor S). Since D′ is optimal, β3 is the only edge thatcross the cycle α0α2β1γ1. By (A), β3 does not cross

neither α0 nor α2, thus, by optimality, β3 only crosseswith µ. Then this case follows from (B).(D.ii) Both R and S contain interior vertices of P .Because Q has no interior vertices of P , at least one β-edge of P goes from R to S. Moreover, such a β-edgeis neither β3 nor βn. Let βi be (i 6= 3, n), the β-edgeof P with less weight. Thus we can get a drawing D∗where the only crossings are those involving βi with µand βi with ν. Since D′ is optimal, cr(D∗) = cr(D′).By (B), we can add the edges α1 and α3 without in-creasing the crossing number.

(E) Some interior vertices of P are in Q.

In this case we have three possibilities.(E.i) All interior vertices of P are in Q. In this casewe can get a drawing with as many crossings as D′in which every crossing involves ρ and some αi, withi = 4, 5, . . . , n. This case follows from (B).(E.ii) Each region R,S and Q contains interior ver-tices of P . Then there must be a β-edge, say βi(respectively βj), crossing the cycle that bounds theregion R (respectively S). By (A), βi (respectivelyβj) must cross either γ1 or β1 (respectively γ2 or β2).If w′(βi) ≤ w′(βj), we can get an optimal drawingD∗ by putting all vertices v4, v5, . . . , vi in S, all ver-tices vi+1, vi+2, . . . , vn in R. Thus we are back incase (D.ii). If w′(βj) ≤ w′(βi) we can proceed analo-gously.(E.iii) BothR andQ contain (all the) interior verticesof P (analogously for S and Q). Let vi be the interiorvertex of P in R with the smallest index. If v4 is in R,then the edge β3 crosses the cycle α0α2β1γ1 and, byputting all interior vertices of P in R, we can proceedas in (D.i). Thus v4 must be in Q and so i ≥ 5. Bythe choice of vi, and (A), βi−1 crosses either γ1 or β1,and all vertices v4, v5, . . . , vi−1 are in Q.It is easy to get a drawing D∗ whose only crossings

are: (i) αj , j = 4, . . . , i− 1 with ρ, (ii) βi−1 crosses µ;and in D∗ all the other vertices vi, . . . , vn remain inR. This implies cr(D∗) = cr(D′) and satisfies (B).

Proof of Theorem 1. Lemma 3 implies that bothα1 and α3 are not crossing-critical in (Gn, w) for anyweight assignment w on Gn. Theorem 1 follows com-bining this and Remark 1.

Finally, consider the following weight assignmentw′n on G′n:

w′n(e) =

1 if e = αi, i = 4, . . . , n;

n− 3 otherwise.

Lemma 4 cr(G′n, w′n) = (n − 3)2 and (G′n, w

′n) is

crossing-critical.

109


Proof. The proof for the crossing number is essen-tially that given in Lemma 3. To prove that the graphis crossing-critical, it is easy to see that for each edgee there exists an optimal drawing of (G′n, w′n) wheree is involved in a crossing.

References[1] L. Beaudou, C. Hernández-Vélez and G. Salazar,

Making a graph crossing-critical by multiplying itsedges, Electron. J. of Combin., 20(1) (2013), #P61.

[2] M. DeVos, B. Mohar and R. Šámal, Unexpected be-havior of crossing sequences, J. Combin. Theory Ser.B 101 (6) (2011), 448–463.

[3] Z. Dvořák and B. Mohar, Crossing-critical graphswith large maximum degree, J. Combin. Theory Ser.B 100 (4) (2010), 413–417.

[4] B. Mohar, R. J. Nowakowski and D. B. West, Re-search problems from the 5th Slovenian Conference(Bled, 2003), Discrete Math. 307(3-5) (2007), 650–658.

110


Witness Bar Visibility

Carmen Cortés∗1, Ferran Hurtado†2, Alberto Márquez‡1, and Jesús Valenzuela§1

1Departamento de Matemática Aplicada I, Universidad de Sevilla.2Departament de Matemàtica Aplicada II, Universitat Politècnica de Catalunya.

Abstract

Bar visibility graphs were introduced in the seventiesas a model for some VLSI layout problems. They havebeen also studied since then by the graph drawingcommunity, and recently several generalizations andrestricted versions have been proposed.We introduce a generalization, witness-bar visibil-

ity graphs, and we prove that this class encompassesall the bar-visibility variations considered so far. Inaddition, we show that many classes of graphs are con-tained in this family, including in particular all planargraphs, interval graphs, circular arc graphs and per-mutation graphs.

1 Introduction and preliminary

denitions

Given a set S of disjoint horizontal line segments inthe plane (called bars hereafter) we say that G is abar-visibility graph if there is a bijection between Sand the vertices of G, and an edge between two ofthese if and only if there is a vertical segment (calledline of sight) between the corresponding bars thatdoes not intersect any other bar. We also say thatS is a bar visibility representation (or a bar visibilitydrawing) of G.Bar visibility graphs were introduced by Garey,

Johnson and So [13] as a modeling tool for digitalcircuit design (see also [16]). These representationsare also a useful tool for displaying diagrams thatconvey visual information on relations among data,which is why many variations of these graphs havebeen considered by the graph drawing community[6, 7, 8, 9, 12, 14, 15].We need some denitions before we can pose pre-

cisely our problem; we use standard terminology asin [5]. We call v-segment any vertical segment. We

∗Email: [email protected].†Email: [email protected]. Research supported

by projects MINECO MTM201230951, Gen. Catalunya

DGR 2009SGR1040, and ESF EUROCORES programme Eu-

roGIGA, CRP ComPoSe: MICINN Project EUI- EURC-2011-

4306, for Spain.‡Email: [email protected].§Email: [email protected].

call ε-segment any axis aligned rectangle having widthε > 0 (intuitively, a thick vertical segment). Let s andt be two horizontal bars. We say that a v-segment con-nects s and t if its endpoints are in s and t. We saythat an ε-segment connects s and t if its horizontalsides are contained in s and t.Let S be a set of non-overlapping horizontal seg-

ments (bars). Two bars s, t ∈ S are visible if, andonly if, there is a v-segment connecting s and t in-tersecting no other segment in S, and we say that sand t are ε-visible if, and only if there is an εsegmentconnecting s and t intersecting no other segment in S.With the preceding denition, bar visibility graphs

as dened in the rst paragraph of this section takeas nodes a set of disjoint bars, and there is an edgebetween two nodes if and only if the correspondingbars are visible (this is also called a strong visibilityrepresentation of the graph [17]). If instead of visibil-ity we require ε-visibility, then we get bar ε-visibilitygraphs or, equivalently, an ε-visibility representationof the graph. The latter have been characterized asthose graphs that admit a planar embedding with allcutpoints on the exterior face [17, 18].A graph G is a weak bar visibility graph if its nodes

can be put in bijection with a set of disjoint bars andthe nodes corresponding to every edge in G are ε-visible (note that not every ε-visibility need be anedge). This family of graphs is exactly the class of allplanar graphs [10].Finally, we say that G is a bar k-visibility graph if

there is a bijection between a set of bars S and thevertices of G, and an edge between two of these if andonly if there is a v-segment joining the correspondingbars that intersects at most k other bars. This gener-alization has been introduced in recent years [8, 12].In this paper we introduce a stronger generaliza-

tion, witness-bar visibility graphs, and we prove thatthis representation approach encompasses all the bar-visibility variations considered so far. In addition, weshow that many classes of graphs are contained inthis family, including in particular all planar graphs,interval graphs, circular arc graphs and permutationgraphs.For the denition of witness-bar visibility graphs

we consider, in addition to the set S of bars that arein correspondence one-to-one with the vertices of the

111


graph being constructed, a set of green bars that fa-vor visibility, and a set of red bars that obstructvisibility. Green bars act as positive witnesses whilered bars correspond to negative witnesses. The barsfrom S neither favor nor obstruct visibilities.For the ease of description it is useful to consider

also purple bars that obstruct visibility in a slightlydierent way than red bars.

Denition 1 Let S, SG, SP and SR be four setsof horizontal segments (bars, green-bars, purple-bars,and red-bars, respectively) such that any two elementsin S ∪ SG ∪ SR are disjoint. We dene:

1. The green-bar visibility graph of S with respectto SG has one vertex for every element in S, andtwo bars s, t ∈ S are adjacent if and only if thereis an ε-segment connecting s and t that crossesat least one green bar.

2. The purple-bar visibility graph of S with respectto SP has one vertex for every element in S, andtwo bars s, t ∈ S are adjacent if and only if thereis an ε-segment connecting s and t that does notcross any purple bar.

3. The witness-bar visibility graph of S with respectto SG and SR has one vertex for every element inS, and two bars s, t ∈ S are adjacent if and onlyif there is an ε-segment connecting s and t thatcrosses strictly more green bars than red bars.

The class of green, purple and witness-bar visibilitygraph are denoted, respectively, by GBG, PBG andWBG.An illustration of the three types of graphs is shown

in Figure 1 (on a black and white printer, node-barsappear as thin lines, red bars as thick dark lines, pur-ple lines as thick lines colored light grey, and the greenlines are seen as thick striped lines).This work is devoted to the study of the classes of

graphs that can be represented via green, purple orbar-visibility graphs and its properties. We start byconsidering the classes GBG and PBG, which will beproved to be subclasses of WBG. Then we will enu-merate classes of graphs that are contained in WBG,as well as properties of this class related to planarity.The terminology witness-bar visibility graphs is in-

spired by the concept of witness proximity graphs,which focuses on deciding neighborliness relationsamong points in a nite set according to the pres-ence of some positive and/or negative witness points,a topic that has been studied in recent years [1, 2, 3,4, 11].

2 The subclasses GBG and PBGIn this section we study the classes GBG and PBGand its relationships with other graph classes. The

a

b

cd

e

xy

z

a

b

cd

e

xy

z

a

b

cd

e

xy

a

b

c d

e

a

b

c d

e

a

b

c d

e

z

Figure 1: Examples of graphs in the families GBG(top), PBG (center) and WBG (bottom).

interest of this is made clear by our rst result:

Lemma 2 The class of graphs WBG contains strictlythe classes GBG and PBG.

An interval graph is the intersection graph of a setof (closed) intervals on the real line.

Theorem 3 Let G be a graph. If G is an intervalgraph, then G ∈ GBG. The reverse is in general false.

The graph class inclusion in Theorem 3 is strict, asone can show that C4, which is not an interval graph,is in GBG. However, graphs with induced cycles oflength greater than 5 are not in GBG.

Proposition 4 If the girth of a graph in GBG is -nite, then it is at most four.

As a consequence of the previous result, it followsthat the green-bar visibility graph class does not con-tain any of the bar-visibility classes described in theintroduction of this paper, because Cn can be repre-sented as weak/ε/strong bar visibilty graph for everyn ≥ 3 [17].Note that even although one may think that the

classes GBG and PBG are related by complementa-tion, possibly by switching purple and green bar col-oring, but it is not the case. For example the union oftwo disjoint triangular cycles is in GBG, as seen in thepreceding section, but its complement is K3,3, whichis not in PBG, a fact that we will see below.On the positive side, let us see that interval graphs

admit a purple-bar visibility representation and provea useful lemma.

112


Theorem 5 If G is an interval graph, then G ∈PBG.

Lemma 6 Let G be a triangle-free graph. If G ∈PBG then G is a planar graph.

Proof. The idea is given in Figure 2.

u1

u2

u3

u4

u5

u6

u7

u8

Figure 2: A purple-bar visibility representation of atriangle-free graph and the corresponding construc-tion of its planar embedding.

Nevertheless the previous result cannot be extendedto a characterization of the class PBG:

Theorem 7 1. K3,3 /∈ PBG.

2. Kn ∈ PBG, ∀n.

3. To admit a purple-bar visibility representation isnot inherited by subgraphs.

Proposition 8 There are nonplanar graphs with tri-angles that do not admit a purple-bar visibility repre-sentation.

An example of these graphs is given in Figure 3.

a

bdc

e

fg h i

j k l m

n

op

Figure 3: A nonplanar graph G with a triangle(gjk), which does not admit a purple-bar visibilityrepresentation.

The class PBG generalizes the classical bar-visibility representations:

Theorem 9 Every graph G that can be representedas strong/ε/weak bar visibility graph admits as well apurple-bar visibility representation.

Corollary 10 Every graph G that can be representedas strong/ε/weak bar visibility graph admits as well awitness-bar visibility representation.

3 The class WBG of witness-bar

visibility graphs

Witness bar visibility also generalizes k-bar visibility:

Theorem 11 Every graph G that can be representedas a bar k-visibility graph admits as well a witness-barvisibility representation.

The idea of the proof is given in Figure 4.

Figure 4: Assume we have to deal with bar 1-visibility,and consider a stack of 5 bars in a strip (left). Wesubdivide the strip into 7 slabs (right). In the secondbar we mimic 1-visibility using WBG-visibility for the3 lowest bars. In the fourth bar we do the same forthe 3 intermediate bars, and in the sixth slab for the3 highest bars.

Theorem 12 The circular arc graphs and permuta-tion graphs are contained in WBG

Given a graph G, let G be the graph resulting fromsubdividing once every edge in G.

Lemma 13 Let G be a graph. If G ∈ WBG then Gis a planar graph.

Lemma 14 K3,3 /∈ WBG and Kc3,3 ∈ WBG.

As a consequence of the preceding lemma we im-mediately obtain the following result:

Theorem 15 The class of the graphs that admit awitness-bar visibility representation is not closed un-der complementation.

We conclude this section with another result on theclass WBG, that discards the possibility of character-izing the class by forbidden minors:

Theorem 16 The property of admitting a witness-bar visibility representation is not inherited by sub-graphs.

Proof. We know that K6 ∈ WBG from Theorem 7and Lemma 2. On the other hand K3,3 is a subdivi-sion of a subgraph ofK6, but we know from Lemma 14that K3,3 is not in WBG. This settles the claim.

113


4 Concluding remarks

Let us summarize the properties we have proved forthe class WBG of witness-bar visibility graphs:

• Every graph G that can be represented asstrong/ε/weak bar visibility graph admits as wella witness-bar visibility representation.

• Every graphG that can be represented as a bar k-visibility graph admits as well a witness-bar vis-ibility representation.

• The class of interval graphs is contained in theclass WBG.

• If G is a circular arc graph or a permutationgraph then G ∈ WBG.

• The class of the graphs that admit a witness-bar visibility representation is not closed undercomplementation.

• The property of admitting a witness-bar visibil-ity representation is not inherited by subgraphs,which discards the possibility of characterizingthe graph class WBG by forbidden minors.

We conclude that the graph class WBG is very richand encompasses many other classes. However, toobtain a characterization or a recognition algorithmappear to be quite challenging problems.

References

[1] O. Aichholzer, R. Fabila, T. Hackl, A.Pilz, P. Ramos, M. van Kreveld, and B.Vogtenhuber. Blocking Delaunay triangu-lations. To appear in Comput. Geom. (ac-cepted 2012). Online version available athttp://dx.doi.org/10.1016/j.comgeo.2012.02.005.

[2] B. Aronov, M. Dulieu and F. Hurtado, Witness(Delaunay) graphs. Comput. Geom. 44(6-7):329-344, 2011.

[3] B. Aronov, M. Dulieu and F. Hurtado, Wit-ness Gabriel graphs. To appear in Comput.Geom. (accepted 2011). Online version at DOI:10.1016/j.comgeo.2011.06.004.

[4] B. Aronov, M. Dulieu, and F. Hurtado, Witnessrectangle graphs. In Algorithms and Data Struc-tures Symposium (WADS), volume 6844 of Lec-tures Notes in Computer Science, pages 73-85.Springer, 2011.

[5] G. Di Battista, P. Eades, R. Tamassia, and I. G.Tollis. Graph Drawing. Prentice Hall Inc., UpperSaddle River, NJ, 1999.

[6] P. Bose, A. M. Dean, J. P. Hutchinson, and T.C. Shermer. On rectangle visibility graphs. In S.C. North, editor, Graph Drawing 1996, volume1190 of Lecture Notes in Comput. Sci., pages 25-44, Berlin, 1997. Springer- Verlag.

[7] G. Chen, J. P. Hutchinson, K. Keating, andJ. Shen. Characterizations of [1,k]-bar visibilitytrees. Electr. J. Comb. 13(1), 2006.

[8] A. M. Dean, W. Evans, E. Gethner, J. D. Lai-son, M. A. Safari, and W. T. Trotter. Bar k-visibility graphs. J. Graph Algorithms & Appli-cations, 11(1):45-59, 2007.

[9] A. M. Dean, E. Gethner, and J. P. Hutchin-son. Unit bar-visibility layouts of triangulatedpolygons: Extended abstract. In J. Pach, edi-tor, Graph Drawing 2004, volume 3383 of LectureNotes in Comput. Sci., pages 111-121, Berlin,2005. Springer-Verlag.

[10] P. Duchet, Y. Hamidoune, M. L. Vergnas, andH. Meyniel. Representing a planar graph by ver-tical lines joining dierent levels. Discrete Math.,46:319-321, 1983.

[11] M. Dulieu, Witness proximity graphs and othergeometric problems, Ph.D. thesis, PolytechnicInstitute of New York University, April 2012.

[12] S. Felsner and M. Massow, Parameters of Bark-Visibility Graphs. J. Graph Algorithms andAppl. vol. 12, no. 1, pp. 5-27 (2008).

[13] M. R. Garey, D. S. Johnson, and H. C. So. Anapplication of graph coloring to printed circuittesting. IEEE Trans. Circuits and Systems, CAS-23(10):591-599, 1976.

[14] J. P. Hutchinson. Arc- and circle-visibilitygraphs. Australas. J. Combin., 25:241-262, 2002.

[15] J. P. Hutchinson, T. Shermer, and A. Vince.On representations of some thicknesstwo graphs.Comput. Geom., 13:161-171, 1999.

[16] M. Schlag, F. Luccio, P. Maestrini, D. Lee, andC. Wong. A visibility problem in VLSI layoutcompaction. In F. Preparata, editor, Advances inComputing Research, volume 2, pages 259-282.JAI Press Inc., Greenwich, CT, 1985.

[17] R. Tamassia and I. G. Tollis. A unied approachto visibility representations of planar graphs. Dis-crete Comput. Geom., 1(4):321-341, 1986.

[18] S. K. Wismath. Characterizing bar line-of-sightgraphs. In Proceedings of the First Symposium ofComputational Geometry, pages 147-152. ACM,1985.

114



Mercè Claverol1, Delia Garijo2, Ferran Hurtado1, Dolores Lara3, and Carlos Seara1

1,3Universitat Politècnica de Catalunya, Barcelona, Spain2Universidad de Sevilla, Spain

Abstract

It is well known that, given n red points and n bluepoints on a circle, it is not always possible to nd aplane geometric Hamiltonian alternating path. In thiswork we prove that if we relax the constraint on thepath from being plane to being 1-plane, then the prob-lem always has a solution, and even a Hamiltonian al-ternating cycle can be obtained on all instances. Wealso extend this kind of result to other congurationsand provide remarks on similar problems.

Introduction

A geometric graph is a graph drawn in the planewhose vertex set is a set of points and whose edgesare straight-line segments connecting pairs of vertices.Two edges of a geometric graph cross if they havean intersection point lying in the relative interior ofboth edges. A plane geometric graph is a geometricgraph without any edge crossings. A 1-plane geomet-ric graph is a geometric graph in which every edgehas at most one crossing. Notice that the terms planegraph and 1-plane graph refer to a geometric object,while to be planar or 1-planar are properties of theunderlying abstract graph. We use here standard no-tation for geometric graphs as in [3] and [12].Let P be a set of points in the plane in general

position (i.e., no three points are collinear), and letCH(P ) denote its convex hull. A geometric spanningtree on P , generically denoted by tree(P ), is any span-ning tree on P whose edges are straight-line segmentsconnecting two points on P . Observe that if |P | = 1,the tree on P is a single vertex. When the tree isa path, we write path(P ). The geometric completegraph K(P ) on P is the complete geometric graphwith vertex set P . Notice that tree(P ) is a spanningtree of K(P ).

1Email: [email protected], [email protected],[email protected]. Partially supported by projectsMINECO MTM2012-30951, Gen. Cat. DGR2009SGR1040,and ESF EUROCORES programme EuroGIGA, CRP Com-PoSe: MICINN Project EUI-EURC-2011-4306.

2Email: [email protected]. Partially supported by projects2009/FQM-164 and 2010/FQM-164.

3Email: [email protected].

Let R and B be two disjoint sets of red and bluepoints in the plane such that no three points of R∪Blie on the same line. The geometric complete bipartitegraph K(R,B) is the graph with vertex set R∪B andwhose edges are all the straight-line segments connect-ing any point in R and any point in B. A line segmentdened by two red points is a red segment, and one de-ned by two blue points is a blue segment. More gen-erally, an edge is said to be monochromatic when thetwo endpoints have the same color, and bichromaticotherwise. The intersections between red segmentsand blue segments are called bichromatic crossings,and those between segments having the same colorare called monochromatic crossings.Problems on adding edges to a given graph to ob-

tain a new graph with some desirable properties are,in general, called augmentation problems. Amongthese, plane augmentation considers an initial planegraph G = (V,E) (possibly empty, i.e., only the pointset V is given) that has to be augmented to anotherplane supergraph G′ = (V,E ∪E′) by adding a set E′

of edges to G, see the survey [7].In this work we focus on problems in which the ini-

tial point set is a bicolored set R ∪ B. This familyof problems has attracted a substantial amount of re-search, see for instance the surveys [8, 7].We rst consider alternating graphs, i.e., those in

which every edge is bichromatic. A well known fact[2, 1, 10, 11] is that given n red points and n bluepoints on a circle (equivalently, in convex position),one cannot always obtain a plane geometric Hamilto-nian alternating path. In this paper we prove that, ifwe relax the constrain on the geometric Hamiltonianalternating path from being plane to being 1-plane,then a solution always exists, even yielding strongerproperties. We also show that the same result holdsfor some other congurations. These results appearin Section 1.Regarding monochromatic graphs, i.e, graphs in

which every edge is monochromatic, it is easy tosee that one cannot always construct a plane perfectmatching in K(R)∪K(B). A trivial example is givenby the vertices of a convex quadrilateral in which twoopposite vertices are colored red and the other two arecolored blue. The same example shows that it is notalways possible to obtain two geometric monochro-

115


matic spanning trees, tree(R) and tree(B), such thattheir union is plane.

The proven nonexistence of plane congurationshad already suggested to researchers to allow somerelaxation in the constraint, but the focus was puton constructing geometric graphs having globally fewcrossings [9, 13]. However, one of the constructions in[13] is in fact a 1-plane graph. We include some re-marks on that particular result and its consequencesin Section 2.

We conclude in Section 3 with some additional com-ments.

1 Alternating graphs: paths and

cycles

In this section we study geometric alternating span-ning graphs on R ∪ B, i.e., spanning subgraphs ofK(R,B). We focus on geometric Hamiltonian alter-nating paths and cycles, which visit red points andblue points alternately. Notice that when there are nocrossings at all, geometric Hamiltonian paths and cy-cles are also called simple polygonals and simple poly-gons, respectively.

1.1 Convex position

The problem of restricting the bicolored point set tolay on a circle has attracted a lot of attention. It iseasy to see that there are sets R and B with the samenumber of points, say n, such that R∪B is in convexposition, and a plane geometric Hamiltonian alter-nating path on R ∪ B cannot exist. Erd®s (see [10])proposed in 1989 to study the value ℓ(n) such thatno matter how the colors are distributed a plane geo-metric alternating path of length at least ℓ(n) alwaysexists. About the same time, Akiyama and Urrutia [2]considered independently the same problem: theyproved a necessary and sucient condition for the ex-istence of a plane geometric Hamiltonian alternatingpath and derived an O(n2) time algorithm to nd one,if it exists. Abellanas et al. [1], and independentlyKyn£l et al. [10], proved that ℓ(n) ≤ 4

3n + O(√n),

and Cibulka et al. [4] showed that ℓ(n) ≥ n+Ω(√n).

These bounds are the best to date. Remind that thetotal number of points is 2n. Also, notice that weslightly abuse the notation by using inequalities, andthat the two bounds have to be read together, notindependently. For more information and details, seeMészáros' PhD Thesis [11].

We next show that if R ∪ B is in convex positionthen a 1-plane geometric Hamiltonian alternating cy-cle not only a path can always be drawn. The fol-lowing lemma is the key tool.

Lemma 1 Let R and B be two disjoint sets of redand blue points in the plane such that R ∪ B is inconvex position, and |R| = |B| = n ≥ 2. Let S be aset of disjoint bichromatic segments on the boundaryof the convex hull of R ∪ B, and |S| = s ≥ 2. Then,there exists a 1-plane geometric Hamiltonian alternat-ing cycle on R ∪ B that contains each segment of Sas an edge.

If R ∪ B is in convex position and |R| = |B| =n ≥ 2 then there are at least two disjoint bichromaticsegments on the boundary of CH(R ∪ B), say s1, s2.Let S = s1, s2. By Lemma 1, one can draw a 1-plane geometric Hamiltonian alternating cycle on R∪B that contains each segment of S as an edge. Thisargument proves our main result in this section.

Theorem 2 Let R and B be two disjoint sets of redand blue points in the plane such that R∪B is in con-vex position, and |R| = |B| = n ≥ 2. Then, there ex-ists a 1-plane geometric Hamiltonian alternating cycleon R ∪B.

1.2 Double chain

We next consider the problem of drawing a 1-planegeometric Hamiltonian alternating cycle on a doublechain whose points are colored red and blue.

The double chain (formally dened below) isa conguration that has been intensively stud-ied since it admits many triangulations, manypolygonizations, many crossing-free matchings, etc.,and for several families of graphs yields themaximum number known to date of such pos-sible congurations among point sets with thesame cardinality (see http://www.cs.tau.ac.il/ shef-fera/counting/PlaneGraphs.html).

A double chain (C1, C2) consists of two oppositeconvex chains C1 and C2, facing each other, such thatthe convex hull of C1∪C2 is a quadrilateral, each pointof C2 lies strictly below every line determined by twopoints of C1, and each point of C1 lies strictly aboveevery line determined by two points of C2. When thepoints of (C1, C2) are colored red and blue, (C1, C2)is said to be a bicolored double chain (see Figure 1),and ri, bi denote the number of red and blue points,respectively, of Ci for i = 1, 2. Note that the sizes ofC1 and C2 may be dierent.

1C

2C

Figure 1: A bicolored double chain (C1, C2).

116


Cibulka et al. [4] proved the following theorem:

Theorem 3 [4] (i) If |Ci| ≥ 15 (|C1| + |C2|) for i =

1, 2, then there exists a non-crossing geometric Hamil-tonian alternating path on (C1, C2); (ii) there exist bi-colored double chains in which one of the chains con-tains at most 1/29 of all the points, which do not ad-mit a non-crossing geometric Hamiltonian alternatingpath.

Theorem 4 below states that when r1+r2 = b1+b2,allowing at most one crossing per edge is strongenough to let us always draw a geometric Hamilto-nian alternating cycle on (C1, C2). The main idea toobtain this cycle is to use Theorem 2 to constructtwo 1-plane geometric alternating cycles Λ1 and Λ2

in CH(C1) and CH(C2), respectively, and to connectthem drawing another 1-plane geometric alternatingcycle Λ in the exterior of CH(C1) and CH(C2). Thisprocess is described next.Let Ei, i = 1, 2, be the set of edges in CH(Ci) that

connect consecutive points of Ci. Suppose rst that3 ≤ b1 + 1 < r1 and 3 ≤ r2 + 1 < b2. Then thereexist at least two monochromatic edges in E1 ∪E2: ared edge rr′ ∈ E1 and a blue edge bb′ ∈ E2. Contractthem obtaining a red point r′′ and a blue point b′′.By Theorem 2, we can draw a cycle Λ1 on the pointset formed by the b1 blue points of C1, the red pointr′′, and b1 − 1 red points of C1 \ r, r′. Analogously,Λ2 is constructed on the point set formed by the r2red points of C2, the blue point b′′, and r2 − 1 bluepoints of C2 \ b, b′. Finally, the cycle Λ is drawn, asin Figure 2, on the remaining r1 − b1 − 1 red pointsof C1, the remaining b2 − r2 − 1 = r1 − b1 − 1 bluepoints of C2, and the points r′′ and b′′. Observe thatr′′ and b′′ connect Λ with Λ1 and Λ2, respectively.By reversing the contraction and deleting the edgesrr′ and bb′, we "open" the three cycles obtaining thedesired cycle on (C1, C2).Note that the preceding argument can easily be

adapted for b1 or r2 equal to zero or one. Observealso that if all the edges in E1 (analogous for E2) arebichromatic then either r1 = b1 (and so r2 = b2) orr1 = b1 + 1 (and b2 = r2 + 1). For these values, evenif there are monochromatic edges in E1, we need touse slightly dierent arguments which are omitted forthe sake of brevity.

Theorem 4 Let R and B be the sets of red andblue points of a bicolored double chain (C1, C2), and|R| = |B| ≥ 2. Then, there exists a 1-plane geometricHamiltonian alternating cycle on (C1, C2).

1.3 General position

The positive results for convex position and for thedouble chain make one wonder whether a similar re-sult holds for any set of points in general position:

Question 1 Let R and B be any two disjoint setsof red and blue points in the plane such that no threepoints of R∪B lie on the same line, and |B| = |R| ≥ 2.Does there always exist a 1-plane geometric Hamilto-nian alternating cycle on R ∪B?

We do not know the answer to the preceding ques-tion. Geometric Hamiltonian cycles with few cross-ings were obtained by Kaneko et al. [9], who gavea tight upper bound of |R| − 1 for the number ofcrossings of a geometric Hamiltonian alternating cy-cle. Figure 2 illustrates a conguration R ∪ B forwhich this upper bound is best possible.

Figure 2: A 1-plane geometric Hamiltonian alternat-ing cycle, on a point set R∪B, with |R|−1 crossings.

To be precise, Kaneko et al. [9] proved the followingresult.

Theorem 5 [9] Let R and B be two disjoint sets ofpoints in the plane such that |R| = |B| and no threepoints of R ∪ B are on the same line. Then we candraw a geometric Hamiltonian alternating cycle onR ∪ B which has at most |R| − 1 crossings. More-over there exist congurations R ∪ B for which thisupper bound |R| − 1 is the best possible.

To prove this theorem, the authors use several lem-mas that we have carefully examined to see whetherit is possible to adapt their proof to obtain a 1-planegraph. However, as far as we can see, there are casesin which the connection of the paths that exist byinduction is not necessarily 1-plane.

It is unclear to us which is the right answer to Ques-tion 1, yet our study leads us to believe that it isnegative in general, yet positive if only a Hamiltonianpath is required, which we state as a conjecture:

Conjecture 1 Let R and B be two disjoint sets of redand blue points in the plane such that no three pointsof R∪B lie on the same line, and |B| ≤ |R| ≤ |B|+1.There always exists a 1-plane geometric Hamiltonianalternating path on R ∪B.

Let us mention that this is ongoing research, cur-rently focusing on the preceding conjecture, which wehope to answer in the next version of this paper.

117


2 A remark on monochromatic

graphs

In this section we would like to remark that a resultobtained by Tokunaga in 1996 can be rephrased interms of 1-plane graphs, hence giving support to theidea that exploring this relaxation may be worth theeort for more problems.On one hand, it is not always possible to construct

a non-crossing perfect matching in K(R) ∪ K(B) asproved by Dumitrescu and Steiger [6]. The originalresult was improved by Dumitrescu and Kaye [5], whoproved that for given R and B, with |R| + |B| = n,there always exists a non-crossing matching inK(R)∪K(B) which covers at least 0.8571 · n points of R ∪B, while for some congurations every non-crossingmatching in K(R) ∪ K(B) covers at most 0.9871 · npoints of R∪B. On the other hand, drawing plane redand blue geometric spanning trees on R and B thatavoid bichromatic crossings is not always possible, andTokunaga [13] characterized their existence in termsof the bichromatic edges on CH(R ∪B).One may wonder whether using 1-plane graphs

would always yield a positive solution to the aboveproblems. The answer is armative: Tokunaga [13]also proved that for given R and B, there exists a pair(path(R), path(B)) of red and blue geometric simpleHamiltonian paths such that each edge of path(R) in-tersects at most one edge of path(B) and vice versa.Having the red and blue geometric simple Hamilto-nian paths from that result, with at most one bichro-matic crossing per edge, we already have got 1-planespanning trees, and taking in each path one segmentout of any two consecutive we get a 1-plane perfectmatching with no monochromatic crossings. This 1-plane matching can also be obtained by using theHam-sandwich theorem and induction on |R ∪B|.We include this remark because the focus in [13] was

to get few crossings rather than achieving the 1-planecharacter, but the consequences show that pursuingthe latter line of research may be of interest.

3 Conclusion

We have proved that several problems on bicoloredpoint sets asking for the construction of a plane geo-metric graph with some requisites, and that have ingeneral negative answer, turn out to have a solution ifthe requirement of the graphs being plane is relaxedto being 1-plane.As mentioned in a previous section, this is ongoing

research and answering Conjecture 1 is our priority.On the other hand, we are also studying the samerelaxation for other problems in which 1-plane graphsmay provide a solution where plane graphs are notsucient.

References

[1] M. Abellanas, A. García, F. Hurtado, and J. Tejel,Caminos alternantes, in: Actas X Encuentros de Ge-

ometría Computacional (in Spanish), 2003, 712.

[2] J. Akiyama and J. Urrutia, Simple alternating pathproblem, Discr. Math. 84 (1990), 101103.

[3] P. Brass, W. Moser, and J. Pach, Research Problems

in Discrete Geometry, Springer, 2005.

[4] J. Cibulka, J. Kyn£l, V. Mészáros, R. Stola°, andP. Valtr, Hamiltonian alternating paths on bicol-ored double-chains, in: Graph Drawing 2008, Lecture

Notes in Computer Science 5417 (2009), 181192.

[5] A. Dumitrescu and R. Kaye, Matching colored pointsin the plane: Some new results, Comput. Geom. The-

ory Appl. 19 (2001), 6985.

[6] A. Dumitrescu and W. Steiger, On a matching prob-lem in the plane, Discr. Math. 211 (2000), 183195.

[7] F. Hurtado and C. D. Tóth, Plane geometric graphaugmentation: a generic perspective, Chapter 16in: Thirty Essays on Geometric Graph Theory (J.Pach, ed.), vol. 29 of Algorithms and Combinatorics,Springer, 2013, 327354.

[8] A. Kaneko and M. Kano, Discrete geometry on redand blue points in the plane - a survey, in: Discreteand Computational Geometry, The Goodman-Pollack

Festschrift; edited by B. Aronov et al., Springer, 2003,551570.

[9] A. Kaneko, M. Kano, and Y. Yoshimoto, AlternatingHamiltonian cycles with minimum number of cross-ings in the plane, Int. J. Comput. Geom. Appl. 10

(2000), 7378.

[10] J. Kyn£l, J. Pach, and G. Tóth, Long alternatingpaths in bicolored point sets, in: Graph Drawing

2004 (J. Pach, ed.), Lecture Notes in Computer Sci-

ence 3383 (2004), 340348. Also in Discr. Math. 308

(2008), 43154322.

[11] V. Mészáros. Extremal problems on planar point setsPhD Thesis, University of Szeged, 2011.

[12] J. Pach, ed. Thirty Essays on Geometric Graph

Theory, vol. 29 of Algorithms and Combinatorics,Springer, 2013.

[13] S. Tokunaga, Intersection number of two connectedgeometric graphs, Information Proc. Discrete Math.

150 (1996), 371378.

118


Phase transitions in the Ramsey-Turán theory

József Balogh∗1

1Department of Mathematics, University of Illinois, Urbana, IL 61801, USA

Abstract

Let f(n) be a function and L be a graph. Denoteby RT(n,L, f(n)) the maximum number of edgesof an L-free graph on n vertices with independencenumber less than f(n). Erd®s and Sós [7] askedif RT (n,K5, c

√n) = o

(n2

)for some constant c.

We answer this question by proving the strongerRT

(n,K5, o

(√n log n

))= o

(n2

). It is known that

RT(n,K5, c

√n log n

)= n2/4 + o

(n2

)for c > 1,

so one can say that K5 has a Ramsey-Turán-phasetransition at c

√n log n. We extend this result to

several other Kp's and functions f(n), determiningmany more phase transitions. We shall formulateseveral open problems, in particular, whether vari-ants of the Bollobás-Erd®s graph, which is a geo-metric construction, exist to give good lower boundson RT (n,Kp, f(n)) for various pairs of p and f(n).These problems are studied in depth by Balogh-Hu-Simonovits [1], where among others, the Szemerédi'sRegularity Lemma and the Hypergraph DependentRandom Choice Lemma are used.

Notation

Denition 1 Denote by RT(n,L,m) the maximumnumber of edges of an L-free graph on n vertices withindependence number less than m.

We are interested in the asymptotic behavior ofRT(n,L, f(n)) and its phase transitions, i.e., in thequestion, when and how the asymptotic behavior ofRT(n,L, f) changes sharply when we replace f by aslightly smaller g.

Denition 2 Let

ρτ(L, f) = lim supn→∞

RT(n,L, f(n))

n2

and

ρτ(L, f) = lim infn→∞

RT(n,L, f(n))

n2.

If ρτ(L, f) = ρτ(L, f), then we write RT(L, f) =ρτ(L, f) = ρτ(L, f), and call RT the Ramsey-Turán

∗Research is partially supported by NSF CAREER GrantDMS-0745185 and Arnold O. Beckman Research Award (UIUCCampus Research Board 13039). [email protected]

density of L with respect to f , ρτ the upper, ρτ thelower Ramsey-Turán densities, respectively.

1 Introduction

Szemerédi [10], using his regularity lemma [11],proved ρτ(K4, o(n)) ≤ 1/8. Bollobás and Erd®s [4]constructed the so-called Bollobás-Erd®s graph, oneof the most important constructions in this area,which shows that ρτ(K4, o(n)) ≥ 1/8. Indeed, theBollobás-Erd®s graph on n vertices is K4-free, with( 18 + o(1))n2 edges and independence number o(n).Later, Erd®s, Hajnal, Sós and Szemerédi [5] extendedthese results, determining RT(K2r, o(n)).

Our focus here is somewhat dierent; we are inter-ested exploring the situation when the independencenumber is really small.

In [7], Erd®s and Sós proved thatRT(n,K5, c

√n) ≤ 1

8n2 + o

(n2

)for every c > 0.

They also asked if RT(n,K5, c√n) = o

(n2

)for

some c > 0. They also suggested the followingconstruction, which with the current state of theart yields RT

(n,K5, c

√n log n

)= n2/4 + o

(n2

)for c > 1: Into each of the parts of a completebalanced bipartite graph place a triangle-free graphwith as small independence number as possible.Then we obtain a K5-free graph with (1/4 + o(1))n2

edges, with low independence number. One of ourmain results answers the question of Erd®s and Sós;practically saying that this construction is optimal.

Theorem 3

RT(n,K5, o

(√n log n

))= o(n2).

We nish this section with an observation, a resultand an open question on K6.

For any c > 1,

ρτ(K6, c

√n log n

)≥ ρτ

(K5, c

√n log n

)≥ 1/4.

(1)

Sudakov [9] using the dependent random choiceproved that

√n is the proper range for the phase-

transition for K6.

119

Ramsey-Turán theory

Theorem 4 RT(K6,

√ne−c

√logn

)= o(n2).

Problem 1 Is ρτ(K6, 0.1

√n log n

)= 0?

Balogh and Lenz [2, 3] using some variants ofthe Bollobás-Erd®s graph solved some longstandingopen problems in this eld. Therefore there is achance that using discrete geometry, someone cansolve Problem 1. Here we describe the Bollobás-Erd®sgraph, and provide the argument for the K5 case.These problems are studied in depth by Balogh-Hu-Simonovits [1], where among others, the Szemerédi'sRegularity Lemma and the Hypergraph DependentRandom Choice Lemma are used.

2 Proof of Theorem 3

The next lemma is taken from the survey of Fox andSudakov [8].

Lemma 5 (Dependent Random Choice Lemma) Leta, d,m, n, r be positive integers. Let G = (V,E) be agraph with n vertices and average degree d = 2e(G)/n.If there is a positive integer t such that

dt

nt−1−(n

r

)(mn

)t

≥ a, (2)

then G contains a subset U of at least a vertices suchthat every r vertices in U have at least m commonneighbors.

We actually prove the following quantitive versionof the theorem.

Theorem 6 For every k > 3, if ω(n) → ∞ then

RT

(n,K5,

√n log n

ω(n)

)≤ n2

(ω(n))1/k= o

(n2

). (3)

Proof. In Theorem 6, we have xed a k > 3 and anω → ∞. If n is large enough, then ε ≥ ω(n)−1/k.Assume that there is a K5-free graph Gn with

e(Gn) ≥ εn2 and α(Gn) <

√n log n

ω(n). (4)

We apply Lemma 5 withn, a = 4n

ω(n)2 , r = 3, d = 2εn, m =√n log n,

and t = k + 3.Now the condition of Lemma 5, (2) is satised as

dt

nt−1−(n

r

)(mn

)t

≥ (2ε)tn− n3

(log n

n

)t/2

> εtn

≥ n

ω(n)1+3/k> a.

So there exists a vertex subset U of G with |U | =a = 4n/ω(n)2 such that all subsets of U of

size 3 have at least m common neighbors. Ei-ther U has an independent set of size at least(

1√2− o(1)

)√4n

ω(n)2 log(

4nω(n)2

)> α(Gn), or Gn[U ]

contains a triangle. In the latter case, denote by Wthe common neighborhood of the vertices of the tri-angle. It follows that |W | ≥ m =

√n log n > α(Gn),

so Gn[W ] contains an edge, and this edge forms a K5

with the triangle.

3 The Bollobás-Erd®s Graph

The Bollobás-Erd®s Graph [4] is a surprising con-struction, which we describe in this section. First welist some properties of the high dimensional sphereSk. Denote µ() the `normalized' measure on Sk,i.e., µ(Sk) = 1. The following properties of the highdimensional sphere is crucial: Given any α, β > 0, itis possible to select ϵ > 0 small enough and then klarge enough so that Properties (P1), (P2), and (P3)below are satised.

(P1) Let C be a spherical cap in Sk with height h,

where 2h =(√

2− ϵ/√k)2

(this means that all

points of the spherical cap are within distance√2− ϵ/

√k of the center). Then µ(C) ≥ 1

2 − α.

(P2) Let C1, . . . , Ct be spherical caps in Sk with height

h, where 2h =(√

2− ϵ/√k)2

. Let zi be the

center of Ci. Assume for all 1 ≤ i < j ≤ t thatd(zi, zj) ≤

√2. Then µ(C1 ∩ . . . ∩ Ct) ≥ 1

2t − tα.

(P3) Let C be a spherical cap with diameter 2 −ϵ/(2

√k). Then µ(C) ≤ β.

We also use the following properties of high dimen-sional spheres.

(P4) For any 0 < γ < 14 , it is impossible to have

p1, p2, q1, q2 ∈ Sk such that d(p1, p2) ≥ 2 − γ,d(q1, q2) ≥ 2 − γ, and d(pi, qj) ≤

√2 − γ for all

1 ≤ i, j ≤ 2.

(P5) Let A ⊆ Sk and let C be a spherical cap of thesame measure. Then diam(A) ≥ diam(C).

(P6) Let A,B ⊆ Sk with equal measure and let C bea cap of the same measure. Then dmax(A,B) ≥diam(C).

Properties (P1) and (P2) follow directly from theformula for the measure of a spherical cap, Proper-ties (P3), (P5), and (P6) are all folklore results thatare easy corollaries of the isoperimetric inequality onthe sphere, and Property (P4) can be proved by ex-amining distances and using the triangle inequality.

120


In order to prove that RT(n,K4, o(n)) ≥ n2

8 , weneed to construct, for every α, β > 0, a K4-free graphG with n vertices, independence number at most βn,

and at least n2

8 (1 − α) edges. Given α, β ≥ 0, takeϵ small enough and k large enough so that Proper-ties (P1) and (P3) hold. Divide the k-dimensionalunit sphere Sk into n/2 domains having equal mea-sure and diameter at most ϵ

10√k. Choose a point from

each domain and let P be the set of these points. Letϕ : P → P(Sk) map points of P to the correspond-ing domain of the sphere. Take as vertex set of Gthe disjoint union of two sets V1 and V2 both isomor-phic to P . For x, y ∈ Vi we make xy an edge of G ifd(x, y) ≥ 2− ϵ/

√k. For x ∈ V1, y ∈ V2 we make xy an

edge of G if d(x, y) ≤√2−ϵ/

√k. Then Property (P1)

shows that every vertex in V1 has at least12 |V2| (1−α)

neighbors in V2 so the total number of edges is at least18n

2(1− α). If I is a set in V1 with |I| ≥ β |V1| = β n2 ,

then µ(ϕ(I)) = |I| / |P | ≥ β. Let C be a spherical capof measure µ(ϕ(I)). Properties (P3) and (P5) showthat 2−ϵ/(2

√k) ≤ diam(C) ≤ diam(ϕ(I)). For p ∈ I,

each ϕ(p) has diameter at most ϵ/(10√k) so we can

nd two points p1, p2 ∈ I with d(p1, p2) ≥ 2 − ϵ/√k,

showing that I is not independent. Finally, Prop-erty (P4) shows this graph has no K4 as a subgraphsince any K4 must take two vertices from V1 and twovertices from V2 (the graph spanned by Vi is triangle-free). To summarize, we have constructed a K4-freegraph G on n vertices with independence number atmost βn and at least 1

8n2(1 − α) edges. Since this

construction holds for any α, β > 0, we have provedthat ρτ(K4, o(n)) ≤ 1/8.

4 Conclusion

The aim of this note was to raise awareness for a prob-lem in extremal graph theory, where a solution couldcome by geometric tools.Acknowledgement: We thank for the anonymous

referee pointing out several typoes of the manuscript.

References

[1] J. Balogh, P. Hu, and M. Simonovits, Phase transi-tions in the Ramsey-Turán theory, manuscript.

[2] J. Balogh and J. Lenz, On the Ramsey-Turán num-bers of graphs and hypergraphs, Israel J. Math. 194

(2013) 4568.

[3] J. Balogh and J. Lenz, Some exact Ramsey-Turánnumbers, Bull. London Math. Soc.

[4] B. Bollobás and P. Erd®s, On a Ramsey-Turántype problem., J. Combinatorial Theory Ser. B,21(2):166168, 1976.

[5] P. Erd®s, A. Hajnal, V. T. Sós, and E. Szemerédi,More results on Ramsey-Turán type problems, Com-

binatorica, 3(1):6981, 1983.

[6] P. Erd®s, A. Hajnal, M. Simonovits, V. T. Sós, andE. Szemerédi, Turán- Ramsey theorems and simpleasymptotically extremal structures. Combinatorica,13 (1):3156, 1993.

[7] P. Erd®s and V. T. Sós, Some remarks on Ramsey'sand Turán's theorem. In Combinatorial theory and itsapplications, II (Proc. Colloq., Balatonfüred, 1969),pages 395404. North-Holland, Amsterdam, 1970.

[8] J. Fox and B. Sudakov. Dependent random choice.Random Structures & Algorithms, 38 (1-2):6899,2011.

[9] B. Sudakov. A few remarks on Ramsey-Turán-typeproblems, J. Combin. Theory Ser. B, 88(1):99106,2003.

[10] E. Szemerédi. On graphs containing no complete sub-graph with 4 vertices, Mat. Lapok, 23:113116, 1973.

[11] E. Szemerédi. Regular partitions of graphs. In Prob-lémes combinatoires et théorie des graphes (Colloq.Internat. CNRS, Univ. Orsay, Orsay, 1976), vol-ume 260 of Colloq. Internat. CNRS, pages 399401.CNRS, Paris, 1978.

121

Ramsey-Turán theory

122



Alfredo García∗1, Clemens Huemer†2, Javier Tejel‡1, and Pavel Valtr§3

1Dept. Métodos Estadísticos. Universidad de Zaragoza. Spain.2Dept. Matemàtica Aplicada IV. Universitat Politècnica Catalunya, Spain.3Department of Applied Mathematics. Charles University, Czech Republic.

Abstract

Given a set S of n points in the plane, in this paper wegive a necessary and sometimes sucient condition tobuild a 4-connected non-crossing geometric graph onS.

Introduction

Given a set S of n points in the plane, a non-crossinggeometric graph on S is a graph in which its verticesare the points of S and its edges are straight-line seg-ments between these points such that no edge passesthrough a vertex dierent from its endpoints and anytwo edges may intersect only at a common endpoint.Since all the geometric graphs considered in this

paper are non-crossing, throughout the paper we willuse the term geometric graph, meaning that the geo-metric graph is non-crossing.The study of geometric graphs and, in particular,

the study of problems on how to embed planar graphsas geometric graphs on given point sets is a very activearea of research (for a review on geometric graphs andsome related topics, see for example [1, 4]). One ofthese problems is the problem of building geometricgraphs with a certain connectivity on a set of pointsS.We will say that a set of points S is k-connectible

if it admits a k-connected geometric graph on it. Fork = 1, 2, 3, it is well-known when S is k-connectibleand how to build a k-connected geometric graph (see[2, 3]). Given S, it is enough to build a non-crossingtree on S (for example the minimum spanning tree of

∗Email: [email protected]. Research partially supported

by projects Gob. Arag. E58-DGA, MINECO MTM2012-30951

and ESF EUROCORES programme EuroGIGA, CRP Com-

PoSe: MICINN Project EUI-EURC-2011-4306.†Email: [email protected]. Research partially sup-

ported by projects MINECO MTM2012-30951 and ESF EU-

ROCORES programme EuroGIGA, CRP ComPoSe: MICINN

Project EUI-EURC-2011-4306.‡Email: [email protected]. Research partially supported by

projects Gob. Arag. E58-DGA, MINECO MTM2012-30951

and ESF EUROCORES programme EuroGIGA, CRP Com-

PoSe: MICINN Project EUI-EURC-2011-4306.§Email: [email protected].

S) when k = 1, and it is enough to build a simplepolygonization of S when k = 2. For the case k = 3,the only set of points not admitting a 3-connectedgeometric graph is the convex case. Otherwise, in [3]the authors give an algorithm to build a 3-connectedgeometric graph using maxd3n/2e, n+m−1 edges,where m is the number of points on the boundary ofthe convex hull of S, and they prove that there is no3-connected plane graph on S with less edges.However, for k > 3, little is known about when a

set of points is k-connectible. For k = 4, Dey et al. [2]show points sets that do not admit any 4-connectedgeometric graph on them and they provide a neces-sary and sucient condition for point sets whose con-vex hull consists of exactly three points. A generalcharacterization of 4- or 5-connectible sets of pointsis not known.In this paper, we study sets of points that are 4-

connectible. We dene a condition (the U-condition)that any set of points must satisfy to be 4-connectibleand we show that the U-condition is always sucientfor some sets of points. By denoting the convex hull ofS by CH(S) and the set of points on the boundary ofthe convex hull of S by H(S), if Q = H(S), I = S \Qand P = H(I), then the U-condition is sucient forsets of points in which Q∪P satises the U-condition.

1 The U-condition

In this section, we will dene the U-condition andwe will see that it is a necessary condition to get 4-connected geometric graphs.A subset C of points of Q is connected if it consists

of consecutive points of Q. We will denote by h(C)the number of connected components of a subset C ofQ.

Denition 1 A set S of points satises the U-condition ifi) |Q| ≤ |I|ii) For any set C ⊂ Q, |H(S \ C)| ≤ |I|+ h(C).

Lemma 2 Every 4-connectible set S satises the U-

condition.

123


Proof. Let us prove that if G(S) is a 4-connectedgraph drawn on S, then S has to satisfy i) and ii).In G(S), each vertex must have degree at least 4, andsince there are no edges linking non-consecutive pointsof Q (otherwise G(S) would not be 3-connected), thenthere are at least 2|Q| edges having an endpoint inQ and the other one in I. On the other hand, asG(S) is 4-connected, each point of I can be linkedto a maximum of two (and consecutive) points of Q.Hence, at most there are 2|I| edges with an endpointin Q and the other one in I. Therefore, 2|Q| ≤ 2|I|and the necessity of i) is proved.Now, suppose C 6= ∅ is a subset of points of Q

and C1, C2, . . . , Ch(C) are its connected components.

Let us denote by Ci the points of Q placed betweencomponent Ci and component Ci+1. Thus, Q consistsof the points C1, C1, C2, C2, . . . , Ch(C), Ch(C) in thisorder. When we remove the points of C, all the pointsin the subsets Ci remain in H(S \ C) and perhaps,between Ci and Ci+1 (mod h(C)), a subset Ii+1 ofpoints of I appears in H(S \C). Either Ii is empty orit consists of consecutive points of P (see Figure 1).Let I be the set I \ (I1 ∪ I2 ∪ . . .∪ Ih(C)) and let C bethe set Q \ C. Observe that proving ii) is equivalentto proving |C| ≤ |I|+ h(C).

C2

C1

I2

I1 q1qkp1

C2

I3

I

C3

C1C3

Figure 1: Illustration of Lemma 2.

Suppose that I1 = p1, . . . , pk′ is nonempty and letq1 and qk be the points of C placed on the boundaryof CH(S \ C) just after and before the points of I1,respectively (see Figure 1). Without loss of generality,we can assume that G(S) is a triangulation. So, eachpoint of I1 must be connected in G(S) to some pointof C1 and, since G(S) is 4-connected, they cannot beconnected to a point of C, except for the edges pk′q1and qkp1. Therefore, for an edge linking a point ofC with an interior point (at least 2|C| edges), theinterior endpoint must be in I \ I1, except for thetwo mentioned edges pk′q1 and qkp1. We can repeatthe same reasoning for every subset Ii, obtaining thatthere must be at least 2|C| − 2h(C) edges with anendpoint in C and the other one in I. On the other

hand, as before, there are at most 2|I| connections ofthis type, so it follows that 2|C| − 2h(C) ≤ 2|I|.

Observe that if |Q| = 3, then |I|+1 = n−2. Hence,part ii) can only fail if we remove one point q of Q andthe boundary of CH(S \ q) contains all the remainingpoints. In [2], this is the condition that is provedto be necessary and sucient to build a 4-connectedgeometric graph (in fact a triangulation) on S.Lastly, let us point out that, given S, checking

whether S satises the U-condition or not can bedone in O(|Q| + |P |) steps, after calculating Q andP . The algorithm is based in the observation (noteasy to prove) that it is not necessary to computeCH(S \C) for all the possible subsets C, but only fora linear number of them.

2 Some 4-connectible sets

In this section, we will give some sets of points forwhich the U-condition is sucient. In particular, wewill see that if Q ∪ P satises the U-condition for aset of points S, then S is 4-connectible. We will useQ (P ) to refer to the convex polygon dened by thepoints of Q (P ).Let us start with the case in which S is precisely

Q ∪ P and |Q| = |P |.

Lemma 3 Let Q = q1, . . . , qn be a set of points in

convex position and let P = p1, . . . , pn be another

set of points in convex position such that P is inside

Q. Suppose that the set of points S = Q ∪ P satises

the U-condition. Then S is 4-connectible.

Proof. Let M be the region CH(Q) \ CH(P ). Toprove the lemma, it is enough to obtain a crossingfree zig-zag cycle Z = piqjpi+1qj+1 . . . pi−1qj−1pi suchthat its edges are in M , because then the edges of Zand the edges of Q and P dene a 4-connected graph(see Figure 2).

P

Q

Z

Figure 2: A 4-connected geometric graph when S =Q ∪ P .

We will say that a triangle qjpipi+1 is legal if itis contained in region M . Proving the lemma isequivalent to proving that there is a sequence of n

124


consecutive legal triangles qjpipi+1, qj+1pi+1pi+2, . . . ,qj+n−1pi+n−1pi+n, where points with equal subscriptsmodulo n are considered identical.

Let us assume that q1, . . . , qm is the set of clock-wise points of Q on the left of the line p1p2. Thefollowing algorithm computes a sequence of n consec-utive legal triangles.

Begin

Do i = j = 2

While (i ≤ n) Do

(* Invariant (1): Triangles of the sequenceqj−1pi−1pi, qj−2pi−2pi−1, . . . , qj−(i−1)p1p2 are legal.*)

If ( Triangle qjpipi+1 is legal ) then

Do i = i+ 1; j = j + 1Else

(* Invariant (2): qj is between qj−1 and the rstcrossing of line pipi+1 with Q *)

Do j = j + 1

End of While

(* After nishing the algorithm, all the trianglesof the sequence qj−np1p2, qj−(n−1)p2p3, . . . , qj−1pnp1are legal.*)

Let us see that assertions (1) and (2) are alwaystrue, so they are invariant in the algorithm.

Trivially, assertion (1) is true the rst time becauseq1p1p2 is legal by hypothesis. Now, suppose that as-sertions (1) and (2) are true in the iterations 1, 2, . . . , kof the loop and let us prove that (1) is still true in thefollowing iteration. If we begin the k+ 1 iteration af-ter exploring a legal triangle in the iteration k, thenclearly (1) is still true (because we are adding the lastexplored triangle to a previous legal sequence). If webegin iteration k+1 after exploring an illegal triangleqjpipi+1 in iteration k, then we need to check thatthe new sequence ST = qjpi−1pi, . . . , qj−i+2p1p2 oftriangles (where j has been increased by one) is legal.Assume to the contrary that in this sequence ST arst illegal triangle qj−hpi−h−1pi−h appears, so qj−his the rst clockwise point of Q on the right of linepi−h−1pi−h. By removing the points of Q from qj+1

to qj−h−1 (see Figure 3 left), then the points of Pfrom pi to pi−h (n−h+1 points) and the points of Qfrom qj−h to qj (h+1 points) appear in the boundaryof the new convex hull, contradicting the U-condition(at most n+ 1 points can appear in the boundary ofthe new convex hull). Therefore (1) is invariant.

For assertion (2), the rst time that the algo-rithm goes to the else branch, we are exploringthe illegal triangle qjpjpj+1, being the trianglesqj−1pj−1pj , . . . , q1p1p2 legal. If qj was on the rightside of pipi+1 and after the second crossing point ofthat line with Q, then, by removing the points of Qfrom q1 to qj−1 (remember that q1 is the rst point ofQ to the left of p1p2), the U-condition is contradictedbecause in the boundary of the new convex hull n+2

P

Q pi

pi+1

pi−1

qj

qj−h pi−h+1

pi−h

pipi+1

qj

qj−1

qj1

pi1

pi1+1

pi1−1

Q1

P

pi−h−1

Q1

Q

Figure 3: Illustration of Lemma 3. Left: Supposingqj−h is on the right of pi−h−1pi−h, the U-conditionfails if Q1 is removed. Right: Supposing qj is on theright of pipi+1, the U-condition fails if Q1 is removed.

points appear (the points of P from p1 to pj+1 andthe points of Q from qj to qn). Therefore, in the rstvisit to the else branch assertion (2) is true.Suppose that the last illegal triangle explored is

qj1pi1pi1+1 and the algorithm is exploring a new il-legal triangle qjpipi+1. As (2) was true in the pre-vious iterations, point qj1 has to be placed betweenqj1−1 and the rst crossing of line pi1pi1+1 with Q.After exploring this triangle, subscript j1 is increasedby one, and then h operations (perhaps h = 0) ofincreasing both subscripts (i and j) are done. There-fore, it must be j = j1 + h + 1 and i = i1 + h,for some h ≥ 0. Since (1) is invariant, the h + 1triangles qj−1pi−1pi, qj−2pi−2pi−1, . . . , qj1pi1−1pi1 arelegal. Therefore, if qj is placed after the second cross-ing of line pipi+1 with Q, then, by removing the hpoints of Q from qj1+1 to qj−1, the h + 2 points ofP from pi1+1 to pi+1 appear in the boundary of thenew convex hull, contradicting the U-condition (seeFigure 3 right). Hence, (2) is invariant.Lastly, since (1) is invariant and the last triangle,

qj−(i−1)p1p2, is legal, then the subscript j − i + 1has to be between 1 and m. This implies thatj ≤ i + m − 1 in the algorithm. Hence, the al-gorithm can go to the else branch a maximum ofm − 1 times, and nishes in a maximum numberof n + m − 1 steps. When the algorithm nishes,then i = n + 1 and the triangles of the sequenceqj−1pnpn+1, qj−2pn−1pn, . . . , qj−np1p2 are legal be-cause (1) is invariant.

Now, assume that |Q| = |P |, Q ∪ P satises theU-condition, there are more points inside P and that|Q| > 3 (the case |Q| = 3 was solved in [2]). To get a4-connected geometric graph, we proceed as follows.First draw the zig-zag including alternatively thepoints of P and Q, according to the previous lemma.Then, take a diagonal of P , for example diagonal p1p3.This diagonal divides P into two subpolygons P1 =p1, p3, . . . , pn, p1 and P ′1 = p1, p2, p3, p1. If the in-terior I(P1) of P1 is nonempty, then let P2 be the con-vex polygon dened by the pointsH(I(P1)∪p1, p3). Ifthe interior I(P2) of this polygon is again nonempty,

125


then we dene P3 as the convex polygon dened bythe points H(I(P2) ∪ p1, p3), and so on, until we ob-tain an empty convex polygon Ph. Thus, we havea sequence Ph ⊂ Ph−1 ⊂ . . . ⊂ P2 ⊂ P1 of nestedpolygons (see Figure 4 left). The same process canbe done starting at P ′1, obtaining another sequenceP ′h′ ⊂ P ′h′−1 ⊂ . . . ⊂ P ′2 ⊂ P ′1 of nested polygons.Observe that the regionMi (M

′i) bounded by the con-

secutive polygons Pi+1 and Pi (P′i+1 and P ′i ) has the

shape of a half-moon. It is not dicult to provethat Mi (M

′i) can be triangulated such that points p1

and p3 are not used, and each one of the added edgeshas an endpoint in Pi (P

′i ) and the other one in Pi+1

(P ′i+1). Then, we triangulate all the half-moons in thisway (using edges connecting points placed in dierentpolygons) and we triangulate the convex polygon Pf ,formed by concatenating Ph and P ′h′ , such that theonly points with degree two are p1 and p3. It can beeasily checked that the triangulation obtained in thisway is 4-connected (see Figure 4 right).

p1

p2

p3

QP1

P2

p1

p2

p3

Q

P ′1

p4

p5

p6p7

p8

Figure 4: The general construction when |Q| = |P |and there are points inside P .

The case in which |Q| < |P | and Q ∪ P satisesthe U-condition is solved in a similar way. First agraph based on a zig-zag is built, although this start-ing graph cannot be a zig-zag as in the previous case,because |Q| < |P |. Now, the starting graph is a zig-zag connecting the points of Q to some points of Pplus some additional edges connecting the points of Pnot belonging to the zig-zag to some points of Q (boldedges in Figure 5 left). After building this startinggraph, we take a diagonal of P connecting two pointsof P , consecutive in the zig-zag but not consecutive inP (points pi1 and pi2 in Figure 5), and we proceed asin the previous case, adding the triangulations of thedierent half-moons and the nal triangulation of theconvex polygon Pf (see Figure 5 right). The resultingtriangulation is 4-connected.The U-condition is the key to nding this starting

graph (the zig-zag plus some additional edges), al-though proving the existence of such a graph is notobvious. Due to space limitations, we do not includethis proof.Therefore, we have proved the following theorem.

Theorem 4 Let S be a set of points. If Q = H(S),P = H(S \ Q) and Q ∪ P satises the U-condition,

Q

pi1pi2

pi3

pi4pi5

P

Figure 5: The general construction when |Q| < |P |and there are points inside P .

then S is 4-connectible.

3 Conclusions

In this paper, we have dened a condition, the U-condition, that any set of points S must satisfy to be4-connectible. Moreover, we have proved that the U-condition is also sucient for sets of points in whichQ ∪ P satises the U-condition.In [2], the case |Q| = 3 is completely solved. Given

a set S of points such that |Q| = 3, the authors showhow to build a 4-connected triangulation on S, exceptfor a particular conguration of points. This particu-lar conguration is precisely the only one not satisfy-ing the U-condition, among all the congurations ofpoints such that |Q| = 3.Using dierent techniques not included in this pa-

per, we can extend the family of 4-connectible sets.For any set S of points satisfying the U-condition (it isnot required that Q∪P satises the U-condition) suchthat |P | = 3 or |P | = 4, we can built a 4-connectedgeometric graph on S.Finally, we conclude with the following conjecture.

Conjecture 1 If a set of points S satises the U-

condition, then S is 4-connectible.

References

[1] P. Brass, W. Moser and J. Pach, Research Problems

in Discrete Geometry, Springer-Verlag, Berlin, 2005.

[2] T.K. Dey, M.B. Dillencourt, S.K. Ghosh and J.M.Cahill, Triangulating with high connectivity, Com-

put. Geom. Theory Appl. 8 (1997), 3956.

[3] A. García, F. Hurtado, C. Huemer, J. Tejel and P.Valtr, On triconnected and cubic plane graphs ongiven point sets, Comput. Geom. Theory Appl. 42

(2009), 913922.

[4] J. Pach (ed.), Thirty Essays on Geometric Graph

Theory, Springer Science+Business Media, NewYork, 2013.

126



Martin Balko∗1, Radoslav Fulek†1, and Jan Kyn£l‡2

1Department of Applied Mathematics, Charles University, Faculty of Mathematics and Physics, Malostranskénám. 25, 118 00 Praha 1, Czech Republic;

2Department of Applied Mathematics and Institute for Theoretical Computer Science, Charles University,Faculty of Mathematics and Physics, Malostranské nám. 25, 118 00 Praha 1, Czech Republic;

Abstract

In 1958, Hill conjectured that the minimum numberof crossings in a drawing of Kn is exactly Z(n) =14b

n2 c⌊n−12

⌋ ⌊n−22

⌋ ⌊n−32

⌋. Generalizing the result by

Ábrego et al. for 2-page book drawings, we provethis conjecture for plane drawings in which edgesare represented by x-monotone curves. In fact, ourproof shows that the conjecture remains true for x-monotone drawings in which adjacent edges do notcross and we count only pairs of edges which crossodd number of times. We also discuss a combinato-rial characterization of these drawings.

1 Introduction

Let G be a graph with no loops and multiple edges.In a drawing D of a graph G in the plane, the verticesare represented by distinct points and each edge isrepresented by a simple continuous arc connecting theimages of its endpoints. As usual, we identify thevertices and their images, as well as the edges and thearcs representing them. It is required that the edgespass through no vertices other than their endpoints.We also assume for simplicity that any two edges haveonly nitely many points in common, no two edgestouch at an interior point and no three edges meet ata common interior point.A crossing in D is a common interior point of two

edges where they properly cross. The crossing numbercr(D) of a drawing D is the number of crossings inD. The crossing number cr(G) of a graph G is the

∗Email: [email protected]. Research supported by thegrant GACR GIG/11/E023 GraDR in the framework of ESFEUROGIGA program, by the Grant Agency of the CharlesUniversity, GAUK 1262213, and by the grant SVV-2013-267313(Discrete Models and Algorithms).†Email: [email protected]. Research supported by

the grant GACR GIG/11/E023 GraDR in the framework ofESF EUROGIGA program.‡Email: [email protected]. Research supported by the

grant GACR GIG/11/E023 GraDR in the framework of ESFEUROGIGA program, by the Grant Agency of the CharlesUniversity, GAUK 1262213, and by the grant SVV-2013-267313(Discrete Models and Algorithms).

minimum of cr(D), taken over all drawings D of G.A drawing D is called simple if no two adjacent edgescross and no two edges have more than one commoncrossing. It is well known and easy to see that everydrawing of G which minimizes the crossing number issimple.According to the famous conjecture of Hill [7, 8]

(also known as Guy's conjecture), the crossing num-ber of the complete graph Kn on n vertices satisescr(Kn) = Z(n), where

Z(n) =1

4

⌊n

2

⌋⌊n− 1

2

⌋⌊n− 2

2

⌋⌊n− 3

2

⌋.

This conjecture has been veried for n ≤ 12 [12] andfor each n, there are drawings of Kn with Z(n) cross-ings [6, 7, 8, 9].A curve α in the plane is x-monotone if every verti-

cal line intersects α in at most one point. A drawingof a graph G in which every edge is represented by anx-monotone curve and no two vertices share the samex-coordinate is called x-monotone (or monotone, forshort). The monotone crossing number mon-cr(G) ofa graph G is the minimum of cr(D), taken over allmonotone drawings D of G.The rectilinear crossing number cr(G) of a graph

G is the smallest number of crossings in a drawingof G where every edge is represented by a straight-line segment. Since every rectilinear drawing of G inwhich no two vertices share the same x-coordinate isx-monotone, we have cr(G) ≤ mon-cr(G) ≤ cr(G) forevery graph G.We call a drawing of a graph semisimple if adja-

cent edges do not cross but independent edges maycross more than once. The monotone semisimple oddcrossing number of G (called monotone odd + bySchaefer [14]), denoted by mon-ocr+(G), is the small-est number of pairs of edges that cross an odd num-ber of times in a monotone semisimple drawing of G.Clearly, mon-ocr+(G) ≤ mon-cr(G).The monotone crossing number has been intro-

duced by Valtr [15] and recently further investigatedby Pach and Tóth [11], who showed that mon-cr(G) <2cr(G)2 holds for every graph G. On the other hand,

127


they showed that the monotone crossing number andthe crossing number are not always the same: thereare graphs G with arbitrarily large crossing numberssuch that mon-cr(G) ≥ 7

6cr(G)− 6.

We study the monotone crossing numbers of com-plete graphs. The drawings of complete graphs withZ(n) crossings obtained by Blaºek and Koman [6](see also [9]) are 2-page book drawings, which maybe considered as a strict subset of x-monotone draw-ings. Thus we have mon-cr(Kn) ≤ Z(n). Ábregoet al. [1] recently proved Hill's conjecture for 2-pagebook drawings of complete graphs. We generalizetheir techniques and show that Hill's conjecture holdsfor all x-monotone drawings of complete graphs, evenfor the monotone semisimple odd crossing number.

Theorem 1 For every n ∈ N, we have

mon-ocr+(Kn) = mon-cr(Kn) = Z(n).

The rectilinear crossing number of Kn is knownto be asymptotically larger than Z(n): this fol-lows from the best current lower bound cr(Kn) ≥(277/729)

(n4

)−O(n3) [3, 5] and from the simple upper

bound Z(n) ≤ 38

(n4

)+O(n3).

See a recent survey by Schaefer [14] for an ency-clopedic treatment of all known variants of crossingnumbers.After submitting this extended abstract, we were

informed that the authors of [1] achieved the resultmon-cr(Kn) = Z(n) already during discussions aftertheir presentation at SoCG 2012, and that it will ap-pear in the proceedings of LAGOS 2013 [2].

2 Monotone Crossing Number

To prove the upper bound on the 2-page crossing num-ber of Kn, Ábrego et al. [1] generalized the notionof k-edges to arbitrary simple drawings of completegraphs. They also introduced the notion of ≤≤k-edges. These capture the essential properties of 2-page book drawings better than ≤k-edges, which hadbeen successfully used before for rectilinear and pseu-dolinear drawings [10, 4, 3]. We show that the ap-proach using ≤≤k-edges can be generalized to arbi-trary semisimple x-monotone drawings.For a semisimple drawing D of Kn and distinct ver-

tices u and v of Kn, let γ be the oriented arc repre-senting the edge u, v. If w is a vertex ofKn dierentfrom u and v, then we say that w is on the left (right)side of γ if the topological triangle uvw with verticesu, v and w traced in this order is oriented counter-clockwise (clockwise, respectively). This generalizesthe denition introduced by Ábrego et al. [1] for sim-ple drawings. However, we were not able to nd ameaningful generalization of this notion to drawings

that are not semisimple, where the edges of the trian-gle uvw can cross several times.A k-edge is an edge u, v of D that has exactly k

points on the same side (left or right). Since everyk-edge has n−2−k points on the other side, every k-edge is also an (n−2−k)-edge and so every edge of Dis a k-edge for some integer k where 0 ≤ k ≤ bn/2c−1.An i-edge with i ≤ k is called a ≤k-edge. Let Ei(D)

be the number of i-edges and E≤k(D) the numberof ≤k-edges of D. Clearly, E≤k(D) =

∑ki=0Ei(D).

Similarly, the number of ≤≤k-edges of D, E≤≤k(D),is dened by the following identity.

E≤≤k(D) =k∑

j=0

E≤j(D) =k∑

i=0

(k + 1− i)Ei(D) (1)

Considering the only three dierent simple draw-ings of K4 up to a homeomorphism of the plane,Ábrego et al. [1] showed that the number of cross-ings in a simple drawing D of Kn can be expressed interms of the number of k-edges in the following way.

Lemma 2 ([1]) For every simple drawing D of Kn

we have

cr(D) = 3

(n

4

)−bn/2c−1∑

k=0

k(n− 2− k)Ek(D), (2)

which can be equivalently rewritten as

cr(D) = 2

bn/2c−2∑k=0

E≤≤k(D)− 1

2

(n

2

)⌊n− 2

2

⌋

− 1

2(1 + (−1)n)E≤≤bn/2c−2(D).

In fact, Lemma 2 can be easily generalized tosemisimple drawings of Kn where cr(D) is replacedby ocr(D), which counts the number of pairs of edgesthat cross an odd number of times in D. The mainreason is that the cycle C4 cannot be drawn in theplane in such a way that both its pairs of oppositeedges cross oddly while adjacent edges do not cross.By Lemma 2, lower bounds on Ek(D) imply lower

bounds on cr(D) and ocr(D). Considering ≤k-edges,Ábrego and Fernández-Merchant [4] and Lovász etal. [10] proved that for rectilinear drawings of Kn,the inequality E≤k ≥ 3

(k+22

)together with (2) gives

cr(G) ≥ Z(n). However, there are simple x-monotone(even 2-page) drawings of Kn where E≤k < 3

(k+22

)for k = 1 [1]. Ábrego et al. [1] showed that similar in-equality for ≤≤k-edges is satised by all 2-page bookdrawings. We show that the same inequality is satis-ed by all x-monotone semisimple drawings of Kn.Let v1, v2, . . . , vn be the vertex set of Kn. Note

that we can assume that all vertices in an x-monotone

128


drawing lie on the x-axis. We also assume that thex-coordinates of the vertices satisfy x(v1) < x(v2) <· · · < x(vn).

Observation 3 Let D be a semisimple drawing ofKn, not necessarily x-monotone. Let v be a vertexincident with the outer face of D and let γi be the ithedge incident with v in the counter-clockwise cyclic or-der such that γ1 and γn−1 are incident with the outerface in a small neighborhood of v. Let vki

be the otherendpoint of γi. Then for every i, j, 1 ≤ i < j ≤ n−1,the triangle vki

vvkjis oriented clockwise. Conse-

quently, for every k, 1 ≤ k ≤ (n − 1)/2, the edgesγk and γn−k are (k − 1)-edges. For even n, the edgeγn/2 is a halving edge.

For an x-monotone drawing D of Kn, we use Ob-servation 3 directly for the vertex vn and then foreach i, for the vertex vi and the subgraph induced byvi, vi+1, . . . , vn.The following denitions were introduced by

Ábrego et al. [1] for 2-page book drawings. Let Dbe a semisimple x-monotone drawing of Kn and letD′ be a drawing obtained from D by deleting the ver-tex vn together with its adjacent edges. A k-edge inD is a (D,D′)-invariant k-edge if it is also a k-edgein D′. It is easy to see that every ≤k-edge in D′ isalso a ≤(k + 1)-edge in D. If 0 ≤ j ≤ k ≤ bn/2c − 1,then a (D,D′)-invariant j-edge is called a (D,D′)-invariant ≤k-edge. Let E≤k(D,D′) denote the num-ber of (D,D′)-invariant ≤k-edges.For i < j, the edge vivj is called the right edge at

vi. The right edges at vi have a natural vertical order.

Lemma 4 Let k be a xed integer such that 0 ≤ k ≤(n−3)/2. For every i ∈ 1, 2, . . . , k+1, the k + 2− ibottommost and the k + 2− i topmost right edges atvi are ≤ k-edges in D. Moreover, at least k + 2− i ofthese ≤ k-edges are (D,D′)-invariant ≤k-edges.

Proof. The rst part of the lemma follows directlyfrom Observation 3. If the edge vivn is one of thek + 2− i topmost right edges at vi, then the k + 2− ibottommost right edges at vi are (D,D′)-invariant≤k-edges. Otherwise the k + 2− i topmost rightedges at vi are (D,D′)-invariant ≤k-edges.

Corollary 5 We have

E≤k(D,D′) ≥k+1∑i=1

(k + 2− i) =

(k + 2

2

).

The following theorem gives the lower bound onthe number of ≤≤k-edges. The proof is essentiallythe same as in [1], we only extracted Lemma 4, whichneeded to be generalized. Together with Lemma 2,Theorem 6 yields Theorem 1.

Theorem 6 Let n ≥ 3 and let D be a semisimple x-monotone drawing of Kn. Then for every k, 0 ≤ k <n/2− 1, we have E≤≤k(D) ≥ 3

(k+33

).

Proof. The proof proceeds by induction on n wherethe case n = 3 is trivially true. Let n ≥ 4 and let Dbe a semisimple x-monotone drawing of Kn. For theinduction step we remove the point vn together withits adjacent edges to obtain a drawing D′ of Kn−1,which is also semisimple and x-monotone.Using Observation 3 we see that for 0 ≤ i ≤ k <

n/2− 1 there are two i-edges adjacent to vn in D andtogether they contribute with 2

∑ki=0(k + 1 − i) =

2(k+22

)to E≤≤k(D) by (1).

Let γ be an i-edge in D′. Then γ contributes by(k− i) to the sum E≤≤k−1(D′) =

∑k−1i=0 (k− i)Ei(D

′).We already observed that γ is either an i-edge or an(i+ 1)-edge in D. If γ is also an i-edge in D (that is,γ is a (D,D′)-invariant i-edge), then it contributes by(k+ 1− i) to E≤≤k(D). This is a gain of +1 towardsE≤≤k−1(D′). If γ is an (i + 1)-edge in D, then itcontributes only (k − i) to E≤≤k(D). Therefore wehave

E≤≤k(D) = 2

(k + 2

2

)+ E≤≤k−1(D′) + E≤k(D,D′).

By the induction hypothesis we know thatE≤≤k−1(D′) ≥ 3

(k+23

)and thus we obtain

E≤≤k(D) ≥ 3

(k + 3

3

)−(k + 2

2

)+ E≤k(D,D′).

The theorem follows by plugging the lower boundfrom Corollary 5.

3 Combinatorial Description

In this section we develop a combinatorial charac-terization of x-monotone drawings which is based onthe signature function introduced by Peters and Szek-eres [13] for describing order types of points sets. LetTn be the set of ordered triples (i, j, k) of the set[n] = 1, 2, . . . , n and let Σn be the set of signaturefunctions σ : Tn → −,+.Let D be an x-monotone drawing of the complete

graph Kn = (V,E) with vertices v1, v2, . . . , vn suchthat their x-coordinates satisfy x(v1) < x(v2) < · · · <x(vn). We assign a signature function σ ∈ Σn to thedrawing D according to the following rule. For eache = vi, vk ∈ E and every integer j, i < j < k,let σ(i, j, k) = − if the point vj lies above the arcrepresenting the edge e and σ(i, j, k) = + otherwise.Note that if the drawing D is also semisimple,

then a triangle vivkvj , j ∈ (i, k), is oriented coun-terclockwise (clockwise) if and only if σ(i, j, k) = −(σ(i, j, k) = +, respectively). It is easy to see that

129


for every signature function σ ∈ Σn there is an x-monotone drawing D which induces σ. However,such a drawing does not have to be semisimple. Weshow a characterization of simple and semisimple x-monotone drawings by small forbidden congurationsin the signature functions.For a, b, c, d ∈ [n] with a < b < c < d and a

signature function σ ∈ Σn we say that the 4-tuple(a, b, c, d) is of the form ξ1ξ2ξ3ξ4 in σ if σ(a, b, c) =ξ1, σ(a, b, d) = ξ2, σ(a, c, d) = ξ3 and σ(b, c, d) = ξ4.

Theorem 7 A signature function σ ∈ Σn can be re-alized by a semisimple x-monotone drawing if andonly if each ordered 4-tuple of indices is of one ofthe forms ++++, −−−−, ++−−, −−++, −++−,+−−+, −−−+, +++−, +−−−, −+++ in σ. Thesignature function σ can be realized by a simple x-monotone drawing if, in addition, there is no 5-tuple(a, b, c, d, e), a < b < c < d < e, with

σ(a, b, e) = σ(a, d, e) = σ(b, c, d) = −σ(a, c, e).

Note that in a simple x-monotone drawing ofKn the crossings can appear only between edgeswhose endpoints induce a 4-tuple of one of the forms++++,−−−−,++−−,−−++,−++−,+−−+.Analogously to a similar correspondence in recti-linear drawings of Kn, we may call these 4-tuplesconvex. Then for a simple x-monotone drawing D ofKn the crossing number of D equals the number ofconvex 4-tuples.A similar notion of convexity for general k-tuples

was used by Peters and Szekeres [13]. This descriptionof crossings is convenient for computer calculations.Using it, we have obtained a complete list of optimalx-monotone drawings of Kn for n ≤ 10.

4 Concluding remarks

It is an interesting direction of further research to seeif similar techniques can be helpful in proving Hill'sconjecture for general drawings of complete graphs.We note that the same approach does not generalizeto all drawings: for example, a particular planar re-alization of the so-called cylindrical drawing [7, 8] ofK10, with crossing number Z(10), does not satisfy thelower bound on ≤≤1-edges in Theorem 6. It wouldalso be interesting to further generalize Theorem 1to monotone drawings where also adjacent edges areallowed to cross.

Acknowledgments

We would like to thank Pavel Valtr for initializing theresearch which led to this problem and Marek Eliá² fordeveloping visualization tools that were helpful duringthe research.

References

[1] B. M. Ábrego, O. Aichholzer, S. Fernández-Merchant,P. Ramos, and G. Salazar, The 2-page crossing num-ber of Kn, preprint, 2012, arXiv:1206.5669.

[2] B. M. Ábrego, O. Aichholzer, S. Fernández-Merchant,P. Ramos, and G. Salazar, More on the crossing num-ber of Kn: Monotone drawings, Proceedings of theVII Latin-American Algorithms, Graphs and Opti-mization Symposium (LAGOS) 2013, Playa del Car-men, Mexico (to appear).

[3] B. M. Ábrego, M. Cetina, S. Fernández-Merchant, J.Leaños, and G. Salazar, On ≤k-edges, crossings, andhalving lines of geometric drawings of Kn, DiscreteComput. Geom. 48 (2012), 192215.

[4] B. M. Ábrego and S. Fernández-Merchant, A lowerbound for the rectilinear crossing number, GraphsCombin. 21 (2005), 293300.

[5] B. M. Ábrego, S. Fernández-Merchant, J. Leaños,and G. Salazar, A central approach to bound thenumber of crossings in a generalized conguration,in: The IVLatin-American Algorithms, Graphs, andOptimization Symposium, Electron. Notes DiscreteMath., 30, Elsevier Sci. B. V., Amsterdam, 2008,273278.

[6] J. Blaºek and M. Koman, A minimal problem con-cerning complete plane graphs, in: Theory of Graphsand its Applications, Proc. Sympos. Smolenice, 1963,Publ. House Czechoslovak Acad. Sci., Prague, 1964,113117.

[7] R. K. Guy, A combinatorial problem, Nabla (Bull.Malayan Math. Soc) 7 (1960), 6872.

[8] F. Harary and A. Hill, On the number of crossingsin a complete graph, Proc. Edinburgh Math. Soc. (2)13 (1963), 333338.

[9] H. Harborth, Special numbers of crossings for com-plete graphs, Discrete Mathematics 244 (2002), 95102.

[10] L. Lovász, K. Vesztergombi, U. Wagner, and E.Welzl, Convex quadrilaterals and k-sets, in: Towardsa theory of geometric graphs, Contemp. Math., 7034,Amer. Math. Soc., Providence, RI, 2004, 139148.

[11] J. Pach and G. Tóth, Monotone Crossing Number,in: Graph drawing, Lecture Notes in Comput. Sci.,7034, Springer, Berlin Heidelberg, 2012, 278289.

[12] S. Pan and B. R. Richter, The crossing number ofK11 is 100, J. Graph Theory 56 (2007), 128134.

[13] G. Szekeres and L. Peters, Computer solution tothe 17-point Erd®s-Szekeres problem, ANZIAM J. 48(2006), 151164.

[14] M. Schaefer, The Graph Crossing Number and itsVariants: A Survey, Electronic Journal of Combina-torics, Dynamic Survey 21 (2013).

[15] P. Valtr, On the pair-crossing number, in: Combina-torial and computational geometry, Math. Sci. Res.Inst. Publ., 52, Cambridge Univ. Press, Cambridge,2005, 569575.

130

http://arxiv.org/abs/1206.5669


Flips in combinatorial pointed pseudo-triangulationswith face degree at most four (extended abstract)

Oswin Aichholzer∗1, Thomas Hackl∗1, David Orden†2, Alexander Pilz∗1, Maria Saumell‡3, andBirgit Vogtenhuber∗1

1Institute for Software Technology, Graz University of Technology, Austria.2Departamento de Física y Matemáticas, Universidad de Alcalá, Spain.3Département d'Informatique, Université Libre de Bruxelles, Belgium.

Abstract

In this paper we consider the ip operation for com-binatorial pointed pseudo-triangulations where faceshave size 3 or 4, so-called combinatorial 4-PPTs. Weshow that every combinatorial 4-PPT is stretchableto a geometric pseudo-triangulation, which in generalis not the case if faces may have size larger than 4.Moreover, we prove that the ip graph of combina-torial 4-PPTs with triangular outer face is connectedand has diameter O(n2).

1 Introduction

Given a graph of a certain class, a ip is the op-eration of removing one edge and inserting a dier-ent one such that the resulting graph is again of thesame class. For the class of maximal planar (sim-ple) graphs, any combinatorial embedding (clockwiseorder of edges around each vertex) has only faces ofsize 3 and hence is called a combinatorial triangu-lation. Flips in combinatorial triangulations removethe common edge of two triangular faces and replaceit by the edge between the two vertices not shared bythe faces, provided that these two vertices where notalready joined by an edge. Combinatorial triangula-tions have a geometric counterpart in triangulationsof point sets in the plane, which are maximal plane

∗Email: [oaich|thackl|apilz|bvogt]@ist.tugraz.at. Researchof OA and BV partially supported by the ESF EUROCORESprogramme EuroGIGA CRP `ComPoSe', Austrian ScienceFund (FWF): I648-N18. Research of TH supported by the Aus-trian Science Fund (FWF): P23629-N18 `Combinatorial Prob-lems on Geometric Graphs'. AP is a recipient of a DOC-fellowship of the Austrian Academy of Sciences.†Email: [email protected]. Research partially supported

by MICINN Project MTM2011-22792, ESF EUROCORESprogramme EuroGIGA - ComPoSe IP04 - MICINN ProjectEUI-EURC-2011-4306 and Junta de Castilla y León ProjectVA172A12-2.‡Email: [email protected]. Research supported

by ESF EuroGIGA project ComPoSe as F.R.S.-FNRS - EU-ROGIGA NR 13604 and by ESF EuroGIGA project GraDR asGAR GIG/11/E023.

geometric (straight-line) graphs with predened ver-tex positions. In this geometric setting there is also aip operation, for which a dierent restriction applies:An edge can be ipped if and only if the two adjacenttriangles form a convex quadrilateral (otherwise thenew edge would create a crossing).Flips in (combinatorial) triangulations have been

thoroughly studied. See [4] for a survey. A promi-nent question about ips is to study the ip graph.This is an abstract graph whose vertices are the mem-bers of the same graph class having the same numberof vertices, and in which two graphs are neighborsif and only if one can be transformed into the otherby a single ip. For both, combinatorial triangula-tions and triangulations (with xed vertex positions),the ip graph is connected. However, the dierentsettings imply linear and quadratic diameter, respec-tively (see [4] for references).Triangulations have a natural generalization in

pseudo-triangulations. They have become a popu-lar structure in Computational Geometry within thelast two decades, with applications in, e.g., rigid-ity theory and motion planning. See [7] for a sur-vey. A pseudo-triangle is a simple polygon in theplane with exactly three convex vertices (i.e., verticeswhose interior angle is smaller than π). A pseudo-triangulation T of a nite point set S in the planeis a partition of the convex hull of S into pseudo-triangles such that the union of the vertices of thepseudo-triangles is exactly S. Triangulations are aparticular type of pseudo-triangulations, actually theones with the maximum number of edges. Thosewith the minimum number of edges are the so-calledpointed pseudo-triangulations, in which every vertex ispointed, i.e., incident to a reex angle (an angle largerthan π).Flips can also be dened for the class of pseudo-

triangulations of point sets in the plane. The ipgraph for general pseudo-triangulations is known tobe connected, as well as the subgraph induced bypointed pseudo-triangulations. The currently bestknown bound on the diameter is O(n log n) for both

131

Flips in combinatorial pointed pseudo-triangulations with face degree at most four (extended abstract)

ip graphs [2, 3].In a pseudo-triangulation, the pseudo-triangles can

have linear size. Hence, in contrast to triangula-tions, the ip operation can no longer be computedin constant time. This fact led to the consideration ofpseudo-triangulations in which the size of the pseudo-triangles is bounded by a constant. Kettner et al. [5]showed that every point set admits a pointed pseudo-triangulation with face degree at most four (except,maybe, for the outer face). We call such a pseudo-triangulation a 4-PPT.On the one hand, 4-PPTs behave nicely for prob-

lems which are hard for general pseudo-triangulations.For instance, they are always properly 3-colorable,while 3-colorability is NP-complete to decide for gen-eral pseudo-triangulations [1]. On the other hand,known properties of general pseudo-triangulations re-main open for 4-PPTs. For instance, it is not knownwhether the ip graph of 4-PPTs is connected, evenfor the basic case of a triangular convex hull.The aim of this paper is to make a step towards

answering this last question, by considering the com-binatorial counterpart of 4-PPTs.A combinatorial pseudo-triangulation [6] is a topo-

logical embedding of a planar simple graph togetherwith an assignment of labels reex/convex to its an-gles such that (1) every interior face has exactly threeangles labeled convex, (2) all the angles of the outerface are labeled reex, and (3) no vertex is incidentto more than one reex angle.Note that this labeling fullls the same proper-

ties as actual reex/convex angles in a (geometric)pseudo-triangulation. This analogy with the geomet-ric case goes on by calling pointed vertices in a com-binatorial pseudo-triangulation those which, indeed,are incident to one angle labeled reex. Then, com-binatorial pointed pseudo-triangulations are those inwhich every vertex is pointed. Combinatorial pointedpseudo-triangulations with face degree at most four(except, maybe, for the outer face), will be called com-binatorial 4-PPTs.

2 Properties

Lemma 1 Let G be a combinatorial 4-PPT and Hbe a subgraph of G with |V (H)| ≥ 3. Then H has atleast 3 vertices whose reex angle is contained in theouter face of H (corners of rst type in [6]).

Corollary 2 In any combinatorial 4-PPT of the in-terior of a simple cycle with b vertices, of which chave the reex angle inside the cycle, the number t oftriangular faces is given by t = b− 2c− 2.

A combinatorial pseudo-triangulation has the gen-eralized Laman property if every subset of x non-poin-ted vertices and y pointed vertices, where x+y ≥ 2,

induces a subgraph with at most 3x + 2y − 3 edges.Both this property and the number of reex anglesfrom Lemma 1 are related to the stretchability of acombinatorial pseudo-triangulation into a geometricone. A face of a combinatorial pseudo-triangulationis called degenerate if it contains edges which appeartwice on the boundary of this face.

Proposition 3 [6, Corollary 2] The following prop-erties are equivalent for a combinatorial pseudo-triangulation G: (1) G can be stretched to becomea pseudo-triangulation. (2) G has the generalizedLaman property. (3) G has no degenerate faces andevery subgraph of G with at least three vertices has atleast three corners of rst type.

Since, by denition, combinatorial 4-PPTs have nodegenerate faces, we can use Proposition 3 to concludethe following.

Theorem 4 Every combinatorial 4-PPT can bestretched to become a 4-PPT with the given assign-ment of angles. Furthermore, combinatorial 4-PPTshave the generalized Laman property.

Note that there exist non-stretchable combinato-rial pointed pseudo-triangulations with faces of sizeat most 5. See Figure 1. There and in the forthcom-ing gures, arcs denote angles labeled as reex.

Figure 1: A non-stretchable combinatorial pointedpseudo-triangulation [6].

3 Flips

In the following we focus on combinatorial 4-PPTswith a xed triangular outer face. For such a com-binatorial 4-PPT, Corollary 2 implies that there isonly one interior triangular face. Before dening ipsbetween combinatorial 4-PPTs, we make some obser-vations about their geometric counterpart.Geometric 4-PPTs with triangular convex hull also

have only one interior triangle. Furthermore, everyedge of the triangle (except for those being part of theconvex hull) is ippable [7]. Observe that the removalof the edge e to be ipped merges the triangle andthe 4-face adjacent at e into a 5-face, which mightbe degenerate if the triangle and the 4-face share twoedges. See Figure 2. Note that this is the only casein which the triangle and the 4-face can share threevertices, as there are no multiple edges in geometricgraphs.

132


Figure 2: Geometric ip of an edge of a triangle. Inthe lower case, removal of the ipped edge gives adegenerate 5-face.

Similar to the geometric case, we consider ips ofan edge e of the (unique) interior triangular face T ina combinatorial 4-PPT (with triangular outer face):Consider the 4-face F sharing e with T . A ip of econsists in replacing e by another edge e′ such that(1) e′ splits (T∪F )\e into a triangular face T ′ and a4-face F ′ and (2) the result is a combinatorial 4-PPT.In particular, and in contrast to the geometric case, inthe combinatorial setting we have to explicitly avoidmultiple edges and thus to ensure that the edge e′

we insert is not already contained in the combinato-rial 4-PPT (as an edge outside T∪F ). The followinglemma shows that every interior edge of the interiortriangular face can be ipped.

Lemma 5 In a combinatorial 4-PPT, every edge eof an interior triangular face that is not an edge ofthe outer face is ippable. Furthermore: (1) If theremoval of e results in a degenerate 5-face, then thereis a unique valid ip for e. (2) If removing e resultsin a non-degenerate 5-face, then there are at least twovalid ips for e.

Observe that, given a combinatorial ip betweentwo combinatorial 4-PPTs, by Theorem 4 we knowthat both of them can be stretched into geometric 4-PPTs with straight edges. However, it might not bepossible to use the same geometric embedding for thevertices in both of them.

4 Flip graph connectivity

Lemma 6 For a given combinatorial 4-PPT with tri-angular outer face and for any edge b of this outerface, there is a sequence of ips resulting in a combi-natorial 4-PPT whose interior triangular face is inci-dent to b.

Once the interior triangular face is incident to anedge b of the outer face, the next step will be ippingaway interior edges incident to one endpoint of b.

Lemma 7 Given a combinatorial 4-PPT with tri-angular outer face, in which the interior triangularface T is incident to the edge b of the outer face, thereis a sequence of ips resulting in a combinatorial 4-PPT in which the endpoint v of b = uv has no interiorincident edges.

Proof. We describe a ip sequence that ips all in-ner edges incident to v. This ip sequence can bepartitioned into two phases and some cases. Let thevertices neighbored to the vertex v be ordered radi-ally around v, starting with u. In each case, let thevertices in that order be u = w0, . . . , wk.

Phase 1: During this phase, the inner triangular faceT has uv as a side, i.e., T = vuw1. We distinguishthree dierent cases:

Case 1: vw1 is the only inner edge incident

to v, i.e., k = 2. If T is incident to only one 4-faceF (i.e., T∪F is degenerate), we can ip the edge vw1

and are done. Otherwise, let the 4-face F incident tovw1 be vw1sw2. See Figure 3. The reex angle insideF is either at s or w1. If it is at s, we ip vw1 to w0s,obtaining the 4-face vw0sw2. Otherwise, the reexangle is at w1 and we ip vw1 to w1w2, obtaining the4-face vw0w1w2. Either way, the degree of v is 2 andwe are done.

v

w1

s

v

w1

w2 w0u = w0

v

w1

v

s

u = w0 w2

w2

w2w0

s

w1s

Figure 3: Phase 1, Case 1: Only one interior edge isincident to v.

Case 2: at least two inner edges are incident

to v and there does not exist an edge w0w2. SeeFigure 4. Since the reex angle of v is at the outer facewe can replace the edge vw1 by w0w2. This reducesthe degree of v by one. The inner triangular face isagain adjacent to w0v, and we remain in Phase 1.

w1

w2

v v

u = w0 w0

w2

w1

Figure 4: Phase 1, Case 2: Several interior edges areincident to v and w0w2 does not exist.

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Flips in combinatorial pointed pseudo-triangulations with face degree at most four (extended abstract)

Case 3: at least two inner edges are incident

to v and there exists an edge w0w2. See Figure 5.If the two inner edges of T are incident to a single4-face, we have a degenerate case; we ip the edgew0w1 to w1w2, making vw1w2 the inner triangularface. Otherwise, let the 4-face F incident to vw1 bevw1sw2; we ip vw1 to vs (this is possible since ifvs already existed, it would have to cross the cycleuw2sw1). Either way, the ip does not reduce thedegree of v, but the inner triangular face is now insidethe 3-cycle vw0w2. We switch to Phase 2.

w1

w2 w2

w2

w1

w2

s s

u = w0

u = w0

w0

w0

v v

v v

w1

w1

Figure 5: Phase 1, Case 3: The possible transitionsto Phase 2.

Phase 2: During this phase, the inner triangular faceis vw1w2, and w1 stays xed for the whole phase.Further, we know that w1 was enclosed by a 3-cycle(at the transition to this phase), which implies thatthere are no edges from w1 to wi for any i ≥ 2. Wedecrease the degree of v in the following manner.Case 1: there is a 4-face F incident to vw2.

There cannot be an edge w1w3 since w1 was enclosedby a 3-cycle. Further, the reex angle of F is notat v. Hence, we can ip vw2 to w1w3, which reducesthe degree of v and we remain in Phase 2, with vw1w3

being the new inner triangular face.Case 2: there is no 4-face incident to vw2, i.e.,

k = 2. This case is symmetric to Case 1 of Phase 1.The edge vw1 is ipped in one of the two describedways, reducing the degree of v to 2 and thus endingthe process.

Theorem 8 The graph of ips in combinatorial 4-PPTs with n vertices and triangular outer face is con-nected and has diameter O(n2).

Proof. Given such a combinatorial 4-PPT, follow thesteps in Lemmas 6 and 7, then use induction for thecombinatorial 4-PPT obtained by removing v. Thisleads to the unique canonical combinatorial 4-PPTwith triangular outer face, where two of the verticesin the outer face are adjacent to all other vertices,while the third one has degree 2. See Figure 6.

Figure 6: A canonical combinatorial 4-PPT.

Furthermore, the number of ips needed in Lemmas 6and 7 is at most linear in the number of vertices ofthe combinatorial 4-PPT.

The presented basic case of combinatorial 4-PPTswith triangular outer face is extendible to an arbi-trarily sized outer face, to labeled vertices, and alsoto the general case of combinatorial 4-PPTs with anarbitrarily sized outer face on labeled vertices. Elabo-rating on these extensions would go beyond the scopeof this extended abstract, though. Details (and omit-ted proofs) can be found in a forthcoming full version.

Acknowledgments. This work was initiated during

the 9th European Research Week on Geometric Graphs

and Pseudo-Triangulations, held May 1418, 2012 in Al-

calá de Henares, Spain. We thank Vincent Pilaud, Pedro

Ramos, and André Schulz for helpful comments.

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[2] O. Aichholzer, F. Aurenhammer, H. Krasser, andP. Braÿ. Pseudotriangulations from surfaces anda novel type of edge ip. SIAM J. Comput.,32(6):16211653, 2003.

[3] S. Bereg. Transforming pseudo-triangulations.Inf. Process. Lett., 90(3):141145, 2004.

[4] P. Bose and F. Hurtado. Flips in planar graphs.Comput. Geom., 42(1):6080, 2009.

[5] L. Kettner, D. Kirkpatrick, A. Mantler,J. Snoeyink, B. Speckmann, and F. Takeuchi.Tight degree bounds for pseudo-triangulations ofpoints. Comput. Geom., 25(12):312, 2003.

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134


Recent developments on the crossing number of the complete graph

Pedro Ramos∗

Department of Physics and Mathematics, University of Alcalá, Alcalá de Henares, Spain.

Introduction

There are two well dened periods in the history of theproblem of nding the crossing number of the com-plete graph. Before 2004, the lack of tools to studythe problem restricted the developments to the studycongurations with a small number of points, the nd-ing of conjectured optimal solutions for arbitrary n,and some fairly basic counting strategies. [4] is anexcellent survey for the main results in this period,including the early history of the problem.In 2004, and independently, Ábrego and Fernández-

Merchant [1] and Lovász et al [5] discovered a verystrong relation between the number of crossings ofa rectilinear drawing of the complete graph and an-other well known object in combinatorial geometry:the number of j-edges of the set of vertices. Thislead to a series of improvements on the known lowerbounds and, although still only one very basic prop-erty of optimal congurations is known (the set of ver-tices has three extreme points), the gap in the leadingterm of the known lower and upper bounds has beengreatly reduced. A very recent survey of all this de-velopments can be found in [2].Recently [3] the concept of j-edge has been gener-

alized to topological drawings of the complete graph,and the relation between crossings and j-edges hasemerged as a promising tool for the general problem.The conjectured bound has already proven to be op-timal for some families of drawings, including 2-pagedrawings and monotone drawings.

References

[1] B. M. Ábrego and S. Fernández-Merchant. A lowerbound for the rectilinear crossing number. Graphs

and Combinatorics, 21:293300, 2005.

[2] B. M. Ábrego and S. Fernández-Merchant, and G.Salazar. The rectilinear crossing number of Kn: Clos-ing in (or are we?). In Thirty essays in geometricgraph theory, J. Pach (Ed), Springer, XVI, pp 504,2013.

∗Email: [email protected]. Partially supported by MEC

grant MTM2011-22792 and by the ESF EUROCORES pro-

gramme EuroGIGA, CRP ComPoSe, under grant EUI-EURC-

2011-4306.

[3] B.M. Ábrego, O. Aichholzer, S. Fernández-Merchant,P. Ramos, and G. Salazar. The 2-page crossing num-ber of Kn. Discrete and Computational Geometry, toappear.

[4] L. Beineke and R. Wilson. The early history of thebrick factory problem. Math. Intelligencer, 32:4148,2010.

[5] L. Lovász, K. Vesztergombi, U. Wagner, and E. Welzl.Convex quadrilaterals and k-sets. In J. Pach, ed-itor, Contemporary Mathematics Series, 342, AMS

2004, volume 342, pp. 139148. American Mathemat-ical Society, 2004.

135

Index of Authors

Abánades, Miguel A.; 77

Aichholzer, Oswin; 81, 131

Bajuelos, Antonio L.; 7

Balko, Martin; 127

Balogh, József; 85, 119

Bereg, Sergey; 3, 65

Botana, Francisco; 77

Cabello, Sergio; 39

Canales, Santiago; 7, 51

Cano, Javier; 91

Chávez, María José; 43

Chimani, Markus; 39

Cibulka, Josef; 103

Claverol, Mercè; 115

Coll, Narcís; 23

Cortés, Carmen; 111

de Miguel, David N.; 69

Díaz-Báñez, José Miguel; 3

Dorzán, Maria Gisela; 27

Enrique, Lluís; 95

Fabila-Monroy, Ruy; 65, 89

Fernández-Fernández, Encarnación; 69

Flores-Peñaloza, David; 65

Fort, Marta; 3, 15, 19

Fulek, Radoslav; 127

Gagliardi, Edilma Olinda; 73

García, Alfredo; 123

Garijo, Delia; 115

González-Aguilar, Hernán; 85

Guerreri, Marité; 23

Hackl, Thomas; 81, 131

Hernández, Gregorio; 7, 27, 51, 73

Hernández-Vélez, César; 107

Hlim¥ný, Petr; 39

Huemer, Clemens; 89, 123

Hurtado, Ferran; 91, 111, 115

Jaume, Rafel; 95

Kirkpatrick, David; 35

Klein, Rolf; 55

Korbelá°, Miroslav; 103

Kyn£l, Jan; 61, 99, 103, 127

Lara, Dolores; 115

Lawrencenko, Serge; 43

Leaños, Jesús; 107

Leguizamón, Mario Guillermo; 27, 73

López, Mario A.; 3, 65

Márquez, Alberto; 111

Martins, Mafalda; 7, 51

Matos, Inês; 7, 51

Mészáros, Viola; 103

Mezura-Montes, Efrén; 27

Orden, David; 69, 131

Pérez-Lantero, Pablo; 3, 65

Pilz, Alexander; 131

Plastria, Frank; 31

Portillo, José R.; 43

Ramos, Pedro; 57, 135

Rodríguez-Nogales, José M.; 69

Sacristán, Vera; 81

Safernová, Zuzana; 61

Salazar, Gelasio; 85

Saumell, Maria; 131

Seara, Carlos; 115

Sellarès, J. Antoni; 15, 19

Steiger, William; 57

Stola°, Rudolf; 103

Tejel, Javier; 123

Tomás, Ana Paula; 11, 47

Tramuns, Eulàlia; 89

Urrutia, Jorge; 1, 3, 91

Valenzuela, Jesús; 111

Valladares, Nacho; 19

Valtr, Pavel; 103, 123

Vila-Crespo, Josena; 69

Villar, M. Trinidad; 43

Vogtenhuber, Birgit; 81, 131

Wallner, Reinhard; 81

Yang, Boting; 35

Zilles, Sandra; 35

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